The document provides an overview of key probability concepts including:
- Sample space is the set of all possible outcomes of a random experiment.
- Mutually exclusive events cannot occur simultaneously.
- Venn diagrams can visually depict relationships between events like intersections.
- Classical probability is the ratio of favorable outcomes to total possible outcomes.
- Relative frequency probability is the limit of observed frequencies of an event over many trials.
- Bayes' theorem relates conditional and inverse conditional probabilities.
Introduction to Discrete Probabilities with Scilab - Michaël Baudin, Consort...Scilab
This document provides an introduction to discrete probabilities with Scilab. It begins with definitions of sets, including union, intersection, complement, difference, and cross product. It then defines discrete distribution functions and probability of events. Properties of probabilities are discussed, such as the probability of a union of disjoint events being the sum of the individual probabilities. The document also covers conditional probability and Bayes' formula. Examples using a six-sided die are provided throughout to illustrate the concepts.
Math 1300: Section 8-3 Conditional Probability, Intersection, and IndependenceJason Aubrey
The document defines conditional probability as the probability of an event occurring given that another event has already occurred. It provides an example of calculating conditional probability using a probability table and the formula P(A|B) = P(A intersect B) / P(B). The document also explains how conditional probability restricts the sample space to outcomes in the given event.
Here the concept of "TRUE" is defined according to Alfred Tarski, and the concept "OCCURING EVENT" is derived from this definition.
From here, we obtain operations on the events and properties of these operations and derive the main properties of the CLASSICAL PROB-ABILITY. PHYSICAL EVENTS are defined as the results of applying these operations to DOT EVENTS.
Next, the 3 + 1 vector of the PROBABILITY CURRENT and the EVENT STATE VECTOR are determined.
The presence in our universe of Planck's constant gives reason to\linebreak presume that our world is in a CONFINED SPACE. In such spaces, functions are presented by Fourier series. These presentations allow formulating the ENTANGLEMENT phenomenon.
Global Journal of Science Frontier Research: FMathematics and Decision Sciences Volume 18 Issue 2 Version 1.0 Year 2018
A New Polynomial-Time Algorithm for Linear ProgrammingSSA KPI
This document summarizes a new polynomial-time algorithm for linear programming.
1) The algorithm reduces the general linear programming problem to a canonical form and solves it through repeated application of projective transformations and optimization over spheres.
2) Each projective transformation followed by optimization reduces the objective function value by a constant factor, allowing the optimal solution to be found in polynomial time.
3) The algorithm runs in O(n3.5L0.5lnLlnlnL) time, an improvement over the ellipsoid method's O(n6L2lnLlnlnL) time.
Here are slides from the presentation Kenny Ascher and I gave at Brown University. The work was part of research done by Caleb Holtzinger, Kenny Ascher, Professor Michael Falk and myself at Northern Arizona University, funded by the National Science Foundation.
It should be noted that an edge-joint partition of a graph is some partition of a graph's edges such that if an edge, e (between vertex a and b), is in a partition then if for any vertex c not equal to a or b, c is in the neighborhood of a AND b then ac and bc are also in the same partition as e. If a graph is simple this is the same as saying that if an edge is in a partition any K_3's that contain this edge are also in said partition.
A graph is edge-joint maximal if there exists no non-trivial edge-joint partition (i.e. the only edge-joint partition is of G with the empty set).
A maximal edge-joint component, A, of a graph, G, is an edge-joint maximal sub-graph, A, such that the partition {E(A), E(G-A)} is an edge-joint partition.
This document provides definitions and properties related to probability theory and statistics. It defines key concepts such as probability spaces, random variables, distribution functions, and probability density functions. It also covers conditional probability, independence, random vectors, and other statistical topics. The document presents the concepts concisely using mathematical notation.
This document discusses several mathematical topics related to analysis, algebra, and geometry. It defines p-adic numbers as the completion of rational numbers with respect to the p-adic valuation. It also defines the p-adic valuation, chain complexes, derived functors, partial differential operators, and Picard's theorems regarding complex functions taking on every value. Additionally, it provides brief definitions for concepts like parabolic subgroups, the Picard variety, and the Picard-Lefschetz theory.
This document discusses probabilistic inference using Bayesian networks and variable elimination. It introduces the concepts of probabilistic inference, Bayesian networks, and variable elimination as a method for performing efficient inference. Variable elimination involves alternating between joining factors and eliminating variables to compute posterior probabilities without enumerating the entire joint distribution. Approximate inference methods like sampling are also discussed as alternatives to exact inference through variable elimination.
Introduction to Discrete Probabilities with Scilab - Michaël Baudin, Consort...Scilab
This document provides an introduction to discrete probabilities with Scilab. It begins with definitions of sets, including union, intersection, complement, difference, and cross product. It then defines discrete distribution functions and probability of events. Properties of probabilities are discussed, such as the probability of a union of disjoint events being the sum of the individual probabilities. The document also covers conditional probability and Bayes' formula. Examples using a six-sided die are provided throughout to illustrate the concepts.
Math 1300: Section 8-3 Conditional Probability, Intersection, and IndependenceJason Aubrey
The document defines conditional probability as the probability of an event occurring given that another event has already occurred. It provides an example of calculating conditional probability using a probability table and the formula P(A|B) = P(A intersect B) / P(B). The document also explains how conditional probability restricts the sample space to outcomes in the given event.
Here the concept of "TRUE" is defined according to Alfred Tarski, and the concept "OCCURING EVENT" is derived from this definition.
From here, we obtain operations on the events and properties of these operations and derive the main properties of the CLASSICAL PROB-ABILITY. PHYSICAL EVENTS are defined as the results of applying these operations to DOT EVENTS.
Next, the 3 + 1 vector of the PROBABILITY CURRENT and the EVENT STATE VECTOR are determined.
The presence in our universe of Planck's constant gives reason to\linebreak presume that our world is in a CONFINED SPACE. In such spaces, functions are presented by Fourier series. These presentations allow formulating the ENTANGLEMENT phenomenon.
Global Journal of Science Frontier Research: FMathematics and Decision Sciences Volume 18 Issue 2 Version 1.0 Year 2018
A New Polynomial-Time Algorithm for Linear ProgrammingSSA KPI
This document summarizes a new polynomial-time algorithm for linear programming.
1) The algorithm reduces the general linear programming problem to a canonical form and solves it through repeated application of projective transformations and optimization over spheres.
2) Each projective transformation followed by optimization reduces the objective function value by a constant factor, allowing the optimal solution to be found in polynomial time.
3) The algorithm runs in O(n3.5L0.5lnLlnlnL) time, an improvement over the ellipsoid method's O(n6L2lnLlnlnL) time.
Here are slides from the presentation Kenny Ascher and I gave at Brown University. The work was part of research done by Caleb Holtzinger, Kenny Ascher, Professor Michael Falk and myself at Northern Arizona University, funded by the National Science Foundation.
It should be noted that an edge-joint partition of a graph is some partition of a graph's edges such that if an edge, e (between vertex a and b), is in a partition then if for any vertex c not equal to a or b, c is in the neighborhood of a AND b then ac and bc are also in the same partition as e. If a graph is simple this is the same as saying that if an edge is in a partition any K_3's that contain this edge are also in said partition.
A graph is edge-joint maximal if there exists no non-trivial edge-joint partition (i.e. the only edge-joint partition is of G with the empty set).
A maximal edge-joint component, A, of a graph, G, is an edge-joint maximal sub-graph, A, such that the partition {E(A), E(G-A)} is an edge-joint partition.
This document provides definitions and properties related to probability theory and statistics. It defines key concepts such as probability spaces, random variables, distribution functions, and probability density functions. It also covers conditional probability, independence, random vectors, and other statistical topics. The document presents the concepts concisely using mathematical notation.
This document discusses several mathematical topics related to analysis, algebra, and geometry. It defines p-adic numbers as the completion of rational numbers with respect to the p-adic valuation. It also defines the p-adic valuation, chain complexes, derived functors, partial differential operators, and Picard's theorems regarding complex functions taking on every value. Additionally, it provides brief definitions for concepts like parabolic subgroups, the Picard variety, and the Picard-Lefschetz theory.
This document discusses probabilistic inference using Bayesian networks and variable elimination. It introduces the concepts of probabilistic inference, Bayesian networks, and variable elimination as a method for performing efficient inference. Variable elimination involves alternating between joining factors and eliminating variables to compute posterior probabilities without enumerating the entire joint distribution. Approximate inference methods like sampling are also discussed as alternatives to exact inference through variable elimination.
Spectral divide-and-conquer algorithms for eigenvalue problems and the SVDyuji_nakatsukasa
The document discusses communication-minimizing algorithms for the symmetric eigendecomposition and singular value decomposition (SVD). Standard algorithms for these problems are expensive in terms of communication cost when reducing matrices to tridiagonal/bidiagonal form. The goal is to design QR-based algorithms that avoid this reduction, thereby minimizing communication. Randomized algorithms provide an alternative approach for approximating the SVD with lower communication cost.
Discussion of Fearnhead and Prangle, RSS< Dec. 14, 2011Christian Robert
The document discusses approximate Bayesian computation (ABC), a technique used when the likelihood function is intractable. ABC works by simulating data under different parameter values and accepting simulations that are close to the observed data according to a distance measure. The key challenges are choosing a sufficient summary statistic of the data and setting the tolerance level. Later sections discuss using a noisy ABC approach, where the summary statistic is perturbed, and calibrating the method so that the ABC posterior converges to the true parameter as the number of simulations increases. The document examines issues around choosing optimal summary statistics and tolerance levels to minimize errors in the ABC approximation.
Workshop on Bayesian Inference for Latent Gaussian Models with ApplicationsChristian Robert
ABC methods provide a way to perform Bayesian inference when the likelihood function is intractable or impossible to compute directly. The basic ABC algorithm works by simulating parameters from the prior and simulating data from those parameters, accepting the parameters if the simulated data is "close" to the actual observed data according to some distance measure and tolerance level.
Later advances include ABC-MCMC which uses an MCMC approach to sample from the posterior, and ABC-NP which adjusts the parameters to better match the observed data rather than rejecting simulations. Other variants such as ABC-SMC and ABC-μ extend the framework to include sequential Monte Carlo methods or jointly model the intractable parameters. Overall, ABC methods provide a
We define an equivalence relation on propositions and a proof system where equivalent propositions have the same proofs. The system obtained this way resembles several known non-deterministic and algebraic lambda-calculi.
This document provides definitions and concepts related to set theory. It begins by defining a set as a collection of well-defined objects or elements. It introduces set notation using capital letters and curly braces. It then defines the cardinality of a set as the number of elements in the set. It discusses subsets, proper subsets, empty sets, finite and infinite sets, universal sets, equivalent sets, and set operations including intersection, union, difference, and complement. It provides examples to illustrate these concepts and introduces Venn diagrams. Finally, it lists laws and theorems related to set operations.
Stinespring’s theorem for maps on hilbert c star moduleswtyru1989
This document discusses Stinespring's theorem for completely positive maps on Hilbert C*-modules. It begins by introducing C*-algebras, Hilbert C*-modules, and completely positive maps. It then presents Stinespring's theorem for completely positive maps between C*-algebras. The document goes on to discuss Asadi's generalization of Stinespring's theorem to completely positive maps between a C*-algebra and bounded operators on a Hilbert space that are compatible with a Hilbert C*-module. It concludes by presenting a further generalization of Stinespring's theorem to completely positive maps between a C*-algebra and a Hilbert C*-module.
Completely positive maps in quantum informationwtyru1989
This document discusses completely positive maps in quantum information. It provides basic notation and definitions related to Hilbert spaces, bounded linear operators, positive semi-definite operators, and C*-algebras. It then defines what a completely positive map is and notes that not all positive maps are completely positive. Stinespring's theorem characterizes completely positive maps and Choi's theorem provides a representation of completely positive maps in terms of matrices. Completely positive maps are important in quantum information as they include quantum channels.
We compute a low-rank surrogate (response surface) approximation to the solution of stochastic PDE. This is a Karhunen-Loeve/polynomial chaos approximation. After that, to compute required statistics, we sample this cheap surrogate, avoiding very expensive solution of the deterministic problem.
Some recent developments in the traffic flow variational formulationGuillaume Costeseque
This document summarizes recent developments in modeling traffic flow using Hamilton-Jacobi equations. It discusses using Hamilton-Jacobi equations to model cumulative vehicle counts on highways with entrance and exit ramps. Source terms are added to the Hamilton-Jacobi equations to account for the effects of exogenous lateral inflows and outflows of vehicles onto the highway. Analytical solutions are presented for cases with constant inflow rates, and for an extended Riemann problem with piecewise constant boundary and inflow conditions.
We propose a way to unify two approaches of non-cloning in quantum lambda-calculi. The first approach is to forbid duplicating variables, while the second is to consider all lambda-terms as algebraic-linear functions. We illustrate this idea by defining a quantum extension of first-order simply-typed lambda-calculus, where the type is linear on superposition, while allows cloning base vectors. In addition, we provide an interpretation of the calculus where superposed types are interpreted as vector spaces and non-superposed types as their basis.
Slides of LNCS 10687:281-293 paper (TPNC 2017). Full paper: https://doi.org/10.1007/978-3-319-71069-3_22
2.2 exponential function and compound interestmath123c
The document discusses exponential functions and their properties. It defines exponential functions as functions of the form f(x) = bx where b > 0 and b ≠ 1. Some key points made in the document include:
- The rules for exponents such as b0, b-k, (√b)k, and (b1/k) are explained.
- Exponential functions are defined for all real numbers x.
- Examples are provided to illustrate calculating exponential expressions and functions with integer, fractional, decimal, and real-number exponents.
- Exponential functions appear in various fields like finance, science, and engineering. Common exponential functions mentioned are y = 10x, y = ex, and y
The dual geometry of Shannon informationFrank Nielsen
The document discusses the dual geometry of Shannon information. It covers:
1. Shannon entropy and related concepts like maximum entropy principle and exponential families.
2. The properties of Kullback-Leibler divergence including its interpretation as a statistical distance and relation to maximum entropy.
3. How maximum likelihood estimation for exponential families can be viewed as minimizing Kullback-Leibler divergence between the empirical distribution and model distribution.
Jan Picek, Martin Schindler, Jan Kyselý, Romana Beranová: Statistical aspects...Jiří Šmída
This document discusses using regression quantiles to estimate time-dependent thresholds for peaks-over-threshold extreme value analysis. It introduces regression quantiles methodology, which allows thresholds to vary based on covariates like time. Exceedances of regression quantile thresholds are shown to follow a generalized Pareto distribution. Tests are developed based on regression rank scores to select appropriate regression models. The approach provides a computationally simple way to incorporate non-stationarity into extreme value analysis.
Classification with mixtures of curved Mahalanobis metricsFrank Nielsen
This document discusses curved Mahalanobis distances in Cayley-Klein geometries and their application to classification. Specifically:
1. It introduces Mahalanobis distances and generalizes them to curved distances in Cayley-Klein geometries, which can model both elliptic and hyperbolic geometries.
2. It describes how to learn these curved Mahalanobis metrics using an adaptation of Large Margin Nearest Neighbors (LMNN) to the elliptic and hyperbolic cases.
3. Experimental results on several datasets show that curved Mahalanobis distances can achieve comparable or better classification accuracy than standard Mahalanobis distances.
AACIMP 2010 Summer School lecture by Leonidas Sakalauskas. "Applied Mathematics" stream. "Stochastic Programming and Applications" course. Part 3.
More info at http://summerschool.ssa.org.ua
This document summarizes Frank Nielsen's talk on divergence-based center clustering and their applications. Some key points:
- Center-based clustering aims to minimize an objective function that assigns data points to their closest cluster centers. This is an NP-hard problem when the number of dimensions and data points are greater than 1.
- Mixed divergences use dual centroids per cluster to define cluster assignments. Total Jensen divergences are proposed as a way to make divergences more robust by incorporating a conformal factor.
- For clustering when centroids do not have closed-form solutions, initialization methods like k-means++ can be used which randomly select initial seeds without computing centroids. Total Jensen k-means++
Yet another statistical analysis of the data of the ‘loophole free’ experime...Richard Gill
I plan to present some simple and as far as I know novel statistical analyses of the data of the famous Bell-type experiments of 2015 and 2016: Delft, NIST, Vienna and Munich.
Every statistical analysis relies on statistical assumptions. I’ll make some quite strong (and obviously naive) assumptions which do however justify a very simple but unconventional analysis, and which enable us to compare the results of the two main types of experiments: the traditional Bell-CHSH type experimental set-up but with settings and state chosen to somehow “optimise” the handling of the detection loophole, and the experiments based on entanglement swapping which do however aim at creating the traditionally optimal state and settings for such experiments.
One cannot say which type of experiment is better without agreeing on how to compromise between the desires to obtain high statistical significance and high physical significance.
I'll also discuss my current opinions on the question: what should we now believe about locality and realism and the foundations of quantum mechanics. My provisional conclusion is "spukhafte Fernwerkung". This is a talk at the 2019 Växjö conference QIRIF
This document provides an overview of Approximate Bayesian Computation (ABC) methods for Bayesian model choice. ABC methods allow Bayesian inference when the likelihood function is intractable or unavailable. The ABC algorithm works by simulating parameters from the prior and accepting simulations where the simulated and observed data are close according to some distance measure and tolerance level. ABC outputs an approximation of the posterior distribution. An example application is presented for choosing a probit model for diabetes risk using data on Pima Indian women.
Control Synthesis by Sum of Squares OptimizationBehzad Samadi
The document outlines a presentation on control synthesis using sum of squares optimization. It begins with an introduction to convex optimization and sum of squares analysis. It then discusses applications of these techniques to control systems and stability analysis. The document provides examples of using sum of squares to solve global optimization problems and verify stability of nonlinear systems.
This document provides an introduction to surds and indices. It discusses different types of numbers including rational and irrational numbers. It explains that surds like the square root of integers are either integers or irrational. The key properties of surds including simplifying expressions with surds are described. Index notation is also introduced as a shorthand for exponents. The basic rules for multiplying and dividing terms with indices are outlined.
The document discusses various topics related to operations management including job design, methods analysis, work measurement, and time studies. It defines job design as specifying the contents and methods of jobs. Methods analysis involves selecting operations to study, documenting current methods, analyzing jobs, proposing and installing improved methods, and follow up. The objectives of time studies are to estimate the time required to perform tasks by timing samples, setting performance standards, and determining allowances for rest breaks.
Spectral divide-and-conquer algorithms for eigenvalue problems and the SVDyuji_nakatsukasa
The document discusses communication-minimizing algorithms for the symmetric eigendecomposition and singular value decomposition (SVD). Standard algorithms for these problems are expensive in terms of communication cost when reducing matrices to tridiagonal/bidiagonal form. The goal is to design QR-based algorithms that avoid this reduction, thereby minimizing communication. Randomized algorithms provide an alternative approach for approximating the SVD with lower communication cost.
Discussion of Fearnhead and Prangle, RSS< Dec. 14, 2011Christian Robert
The document discusses approximate Bayesian computation (ABC), a technique used when the likelihood function is intractable. ABC works by simulating data under different parameter values and accepting simulations that are close to the observed data according to a distance measure. The key challenges are choosing a sufficient summary statistic of the data and setting the tolerance level. Later sections discuss using a noisy ABC approach, where the summary statistic is perturbed, and calibrating the method so that the ABC posterior converges to the true parameter as the number of simulations increases. The document examines issues around choosing optimal summary statistics and tolerance levels to minimize errors in the ABC approximation.
Workshop on Bayesian Inference for Latent Gaussian Models with ApplicationsChristian Robert
ABC methods provide a way to perform Bayesian inference when the likelihood function is intractable or impossible to compute directly. The basic ABC algorithm works by simulating parameters from the prior and simulating data from those parameters, accepting the parameters if the simulated data is "close" to the actual observed data according to some distance measure and tolerance level.
Later advances include ABC-MCMC which uses an MCMC approach to sample from the posterior, and ABC-NP which adjusts the parameters to better match the observed data rather than rejecting simulations. Other variants such as ABC-SMC and ABC-μ extend the framework to include sequential Monte Carlo methods or jointly model the intractable parameters. Overall, ABC methods provide a
We define an equivalence relation on propositions and a proof system where equivalent propositions have the same proofs. The system obtained this way resembles several known non-deterministic and algebraic lambda-calculi.
This document provides definitions and concepts related to set theory. It begins by defining a set as a collection of well-defined objects or elements. It introduces set notation using capital letters and curly braces. It then defines the cardinality of a set as the number of elements in the set. It discusses subsets, proper subsets, empty sets, finite and infinite sets, universal sets, equivalent sets, and set operations including intersection, union, difference, and complement. It provides examples to illustrate these concepts and introduces Venn diagrams. Finally, it lists laws and theorems related to set operations.
Stinespring’s theorem for maps on hilbert c star moduleswtyru1989
This document discusses Stinespring's theorem for completely positive maps on Hilbert C*-modules. It begins by introducing C*-algebras, Hilbert C*-modules, and completely positive maps. It then presents Stinespring's theorem for completely positive maps between C*-algebras. The document goes on to discuss Asadi's generalization of Stinespring's theorem to completely positive maps between a C*-algebra and bounded operators on a Hilbert space that are compatible with a Hilbert C*-module. It concludes by presenting a further generalization of Stinespring's theorem to completely positive maps between a C*-algebra and a Hilbert C*-module.
Completely positive maps in quantum informationwtyru1989
This document discusses completely positive maps in quantum information. It provides basic notation and definitions related to Hilbert spaces, bounded linear operators, positive semi-definite operators, and C*-algebras. It then defines what a completely positive map is and notes that not all positive maps are completely positive. Stinespring's theorem characterizes completely positive maps and Choi's theorem provides a representation of completely positive maps in terms of matrices. Completely positive maps are important in quantum information as they include quantum channels.
We compute a low-rank surrogate (response surface) approximation to the solution of stochastic PDE. This is a Karhunen-Loeve/polynomial chaos approximation. After that, to compute required statistics, we sample this cheap surrogate, avoiding very expensive solution of the deterministic problem.
Some recent developments in the traffic flow variational formulationGuillaume Costeseque
This document summarizes recent developments in modeling traffic flow using Hamilton-Jacobi equations. It discusses using Hamilton-Jacobi equations to model cumulative vehicle counts on highways with entrance and exit ramps. Source terms are added to the Hamilton-Jacobi equations to account for the effects of exogenous lateral inflows and outflows of vehicles onto the highway. Analytical solutions are presented for cases with constant inflow rates, and for an extended Riemann problem with piecewise constant boundary and inflow conditions.
We propose a way to unify two approaches of non-cloning in quantum lambda-calculi. The first approach is to forbid duplicating variables, while the second is to consider all lambda-terms as algebraic-linear functions. We illustrate this idea by defining a quantum extension of first-order simply-typed lambda-calculus, where the type is linear on superposition, while allows cloning base vectors. In addition, we provide an interpretation of the calculus where superposed types are interpreted as vector spaces and non-superposed types as their basis.
Slides of LNCS 10687:281-293 paper (TPNC 2017). Full paper: https://doi.org/10.1007/978-3-319-71069-3_22
2.2 exponential function and compound interestmath123c
The document discusses exponential functions and their properties. It defines exponential functions as functions of the form f(x) = bx where b > 0 and b ≠ 1. Some key points made in the document include:
- The rules for exponents such as b0, b-k, (√b)k, and (b1/k) are explained.
- Exponential functions are defined for all real numbers x.
- Examples are provided to illustrate calculating exponential expressions and functions with integer, fractional, decimal, and real-number exponents.
- Exponential functions appear in various fields like finance, science, and engineering. Common exponential functions mentioned are y = 10x, y = ex, and y
The dual geometry of Shannon informationFrank Nielsen
The document discusses the dual geometry of Shannon information. It covers:
1. Shannon entropy and related concepts like maximum entropy principle and exponential families.
2. The properties of Kullback-Leibler divergence including its interpretation as a statistical distance and relation to maximum entropy.
3. How maximum likelihood estimation for exponential families can be viewed as minimizing Kullback-Leibler divergence between the empirical distribution and model distribution.
Jan Picek, Martin Schindler, Jan Kyselý, Romana Beranová: Statistical aspects...Jiří Šmída
This document discusses using regression quantiles to estimate time-dependent thresholds for peaks-over-threshold extreme value analysis. It introduces regression quantiles methodology, which allows thresholds to vary based on covariates like time. Exceedances of regression quantile thresholds are shown to follow a generalized Pareto distribution. Tests are developed based on regression rank scores to select appropriate regression models. The approach provides a computationally simple way to incorporate non-stationarity into extreme value analysis.
Classification with mixtures of curved Mahalanobis metricsFrank Nielsen
This document discusses curved Mahalanobis distances in Cayley-Klein geometries and their application to classification. Specifically:
1. It introduces Mahalanobis distances and generalizes them to curved distances in Cayley-Klein geometries, which can model both elliptic and hyperbolic geometries.
2. It describes how to learn these curved Mahalanobis metrics using an adaptation of Large Margin Nearest Neighbors (LMNN) to the elliptic and hyperbolic cases.
3. Experimental results on several datasets show that curved Mahalanobis distances can achieve comparable or better classification accuracy than standard Mahalanobis distances.
AACIMP 2010 Summer School lecture by Leonidas Sakalauskas. "Applied Mathematics" stream. "Stochastic Programming and Applications" course. Part 3.
More info at http://summerschool.ssa.org.ua
This document summarizes Frank Nielsen's talk on divergence-based center clustering and their applications. Some key points:
- Center-based clustering aims to minimize an objective function that assigns data points to their closest cluster centers. This is an NP-hard problem when the number of dimensions and data points are greater than 1.
- Mixed divergences use dual centroids per cluster to define cluster assignments. Total Jensen divergences are proposed as a way to make divergences more robust by incorporating a conformal factor.
- For clustering when centroids do not have closed-form solutions, initialization methods like k-means++ can be used which randomly select initial seeds without computing centroids. Total Jensen k-means++
Yet another statistical analysis of the data of the ‘loophole free’ experime...Richard Gill
I plan to present some simple and as far as I know novel statistical analyses of the data of the famous Bell-type experiments of 2015 and 2016: Delft, NIST, Vienna and Munich.
Every statistical analysis relies on statistical assumptions. I’ll make some quite strong (and obviously naive) assumptions which do however justify a very simple but unconventional analysis, and which enable us to compare the results of the two main types of experiments: the traditional Bell-CHSH type experimental set-up but with settings and state chosen to somehow “optimise” the handling of the detection loophole, and the experiments based on entanglement swapping which do however aim at creating the traditionally optimal state and settings for such experiments.
One cannot say which type of experiment is better without agreeing on how to compromise between the desires to obtain high statistical significance and high physical significance.
I'll also discuss my current opinions on the question: what should we now believe about locality and realism and the foundations of quantum mechanics. My provisional conclusion is "spukhafte Fernwerkung". This is a talk at the 2019 Växjö conference QIRIF
This document provides an overview of Approximate Bayesian Computation (ABC) methods for Bayesian model choice. ABC methods allow Bayesian inference when the likelihood function is intractable or unavailable. The ABC algorithm works by simulating parameters from the prior and accepting simulations where the simulated and observed data are close according to some distance measure and tolerance level. ABC outputs an approximation of the posterior distribution. An example application is presented for choosing a probit model for diabetes risk using data on Pima Indian women.
Control Synthesis by Sum of Squares OptimizationBehzad Samadi
The document outlines a presentation on control synthesis using sum of squares optimization. It begins with an introduction to convex optimization and sum of squares analysis. It then discusses applications of these techniques to control systems and stability analysis. The document provides examples of using sum of squares to solve global optimization problems and verify stability of nonlinear systems.
This document provides an introduction to surds and indices. It discusses different types of numbers including rational and irrational numbers. It explains that surds like the square root of integers are either integers or irrational. The key properties of surds including simplifying expressions with surds are described. Index notation is also introduced as a shorthand for exponents. The basic rules for multiplying and dividing terms with indices are outlined.
The document discusses various topics related to operations management including job design, methods analysis, work measurement, and time studies. It defines job design as specifying the contents and methods of jobs. Methods analysis involves selecting operations to study, documenting current methods, analyzing jobs, proposing and installing improved methods, and follow up. The objectives of time studies are to estimate the time required to perform tasks by timing samples, setting performance standards, and determining allowances for rest breaks.
This document summarizes an Export Quotation Worksheet form used by companies to verify and consider all costs of an export transaction. The form addresses export costs that may be paid by the seller or included in the invoice to the buyer according to the sale terms. These costs include packing, inland freight, freight by transport mode, insurance, and special costs like bank and consular fees. The form also allows comparing up to three quotes for packing and freight charges and can serve as a worksheet for preparing a Pro Forma Invoice.
The document describes an HR portal that provides employees, new hires, recruits, and retirees access to important HR information and services. The portal offers sections for benefits, career opportunities, policies, training, and other resources. It allows employees to manage their benefits and information, conduct surveys, view pay statements, and submit expense claims. HR staff can upload documents, send communications, and manage administrative functions and user permissions through the portal. The portal provides a centralized system to disseminate information and conduct HR activities efficiently.
This document discusses discrete probability distributions, specifically the binomial and Poisson distributions. It provides information on calculating probabilities using the binomial and Poisson probability formulas and tables. It defines key characteristics of binomial experiments and conditions for applying the binomial and Poisson distributions. Examples are given to demonstrate calculating probabilities for each distribution, including finding the mean, variance and standard deviation for binomial distributions.
The document provides steps to calculate the mean, variance, and standard deviation of a probability distribution. It defines a probability distribution with values from 2 to 10 and probabilities from 0.15 to 0.35. It then calculates the mean as the sum of x*P(x), creates columns for x^2 and x^2*P(x) to calculate variance, and finds variance as the sum of x^2*P(x) - mean^2.
Calculate Standard Deviation
https://www.easycalculation.com/statistics/standard-deviation.php
Online calculator to find the mean, variance and standard deviation from a set of given data.
The document provides information on the functions and structure of the Reserve Bank of India (RBI). It summarizes that RBI was established in 1935 and is now owned by the central government. Its main functions include acting as a bank for banks and the government, managing currency and foreign exchange, conducting monetary policy, regulating and supervising banks, and promoting development. Internally, RBI is headed by a governor and organized into departments that handle functions like currency, banking, supervision and policy.
Variance and standard deviation of a discrete random variableccooking
The document shows the steps to calculate the variance and standard deviation of a probability distribution. It involves creating columns for the random variable x, the probability P(x), the products x*P(x) and x^2*P(x). The mean is calculated as the sum of x*P(x). The variance is calculated as the sum of x^2*P(x) - the mean squared.
The document is an advertisement for NAMO, a notable advertising media organization that specializes in indoor advertising. Some key points:
- NAMO utilizes internet-driven indoor displays like billboards, display boards, and TVs to deliver cost-effective advertising that targets specific audiences.
- Indoor advertising is effective as it receives undivided attention in a relaxed environment for 1-2 minutes, with high ad recall and memorization.
- NAMO offers various advertising packages on indoor displays located in businesses across different locations in India, with pricing options for monthly, 6-month, or annual plans.
- Features include displaying static or video ads, proof of performance tracking, and options for coupons
1) The document provides 11 examples involving calculations using the normal distribution to solve probability problems related to business, quality control, and sampling.
2) Many of the examples ask the reader to calculate the probability of an event occurring or the expected number of outcomes given data about the average and standard deviation of a normal distribution.
3) The final example discusses whether a concession manager should hire additional employees given the potential costs and probabilities associated with the expected attendance at a hockey game.
The document discusses elementary theorems and concepts related to probability and conditional probability. It defines the addition rule for mutually exclusive events, the formula for calculating probability of an event as the sum of probabilities of individual outcomes, and the general addition rule for probability. It also defines conditional probability as the probability of an event A given that another event B has occurred, and introduces Bayes' theorem which provides a formula for calculating the probability of an event given certain conditions.
The document discusses elementary theorems and concepts related to conditional probability, including:
1. Theorems for calculating the probability of unions and intersections of events.
2. The definition of conditional probability as the probability of an event A given that another event B has occurred.
3. Bayes' theorem, which provides a formula for calculating the probability of an event A given event B in terms of probabilities of events B given A.
The document defines key concepts in probability theory including experiments, outcomes, sample spaces, events, operations on events like union and intersection, and properties of events like mutual exclusiveness and collective exhaustiveness. It also covers definitions and properties of probability, including relative frequency and axioms of probability. Additional concepts summarized are conditional probability, total probability theorem, independent events, and Bayes' theorem.
The document provides information about probability and statistics concepts including:
1) Mathematical, statistical, and axiomatic definitions of probability are given along with examples of mutually exclusive, equally likely, and independent events.
2) Laws of probability such as addition law, multiplication law, and total probability theorem are defined and formulas are provided.
3) Concepts of random variables, discrete and continuous random variables, probability mass functions, probability density functions, and expected value are introduced.
This document provides an overview of key concepts in probability and statistics, including:
- Definitions of probability, sample spaces, events, and the axioms of probability
- Concepts of conditional probability, Bayes' rule, independence, and discrete random variables
- How to calculate probabilities of events, expected values, variance, and conditioning probabilities on other events or random variables
Show that if A is a fixed event of positive probability, then the fu.pdfakshitent
Show that if A is a fixed event of positive probability, then the function Q[B]=P[B|A] taking
events B into R satisfies the three defining axioms of probability.
Here is the three defining axioms of probability.
A probability measure P is a function taking the family of events H to the real numbers such that
(i) P[Pi]=1
(ii) For all A includes in H, P[A] 0.
(iii) If A1, A2,.....is a sequence of pairwise disjoint events then
P[A1 U A2 U.....]=P[Ai]
Solution
say A1 , A2 , . . ., are called mutually disjoint or pairwise disjoint if Ai n A j = 0 for
any two of the events Ai and A j ; that is, no two of the events overlap. According to
Kolmogorov’s axioms, each event A has a probability P(A), which is a number. These numbers
satisfy three axioms: Axiom 1: For any event A, we have P(A) = 0. Axiom 2: P(S ) = 1. 4
Axiom 3: If the events A1 , A2 , . . . are pairwise disjoint, then CHAPTER 1. BASIC IDEAS
P(A1 ? A2 ? · · ·) = P(A1 ) + P(A2 ) + · · · Note that in Axiom 3, we have the union of events
and the sum of numbers. Don’t mix these up; never write P(A1 ) ? P(A2 ), for example.
Sometimes we sep- arate Axiom 3 into two parts: Axiom 3a if there are only ?nitely many events
A1 , A2 , . . . , An , so that we have P(A1 ? · · · ? An ) = ? P(Ai ), n i=1 and Axiom 3b for
in?nitely many. We will only use Axiom 3a, but 3b is important later on. Notice that we write n
? P(Ai) i=1 for P(A1 ) + P(A2 ) + · · · + P(An ). 1.4 Proving things from the axioms You can
prove simple properties of probability from the axioms. That means, every step must be justi?ed
by appealing to an axiom. These properties seem obvious, just as obvious as the axioms; but the
point of this game is that we assume only the axioms, and build everything else from that. Here
are some examples of things proved from the axioms. There is really no difference between a
theorem, a proposition, and a corollary; they all have to be proved. Usually, a theorem is a big,
important statement; a proposition a rather smaller statement; and a corollary is something that
follows quite easily from a theorem or proposition that came before. Proposition 1.1 If the event
A contains only a ?nite number of outcomes, say A = {a1 , a2 , . . . , an }, then P(A) = P(a1 ) +
P(a2 ) + · · · + P(an ). To prove the proposition, we de?ne a new event Ai containing only the
out- come ai , that is, Ai = {ai }, for i = 1, . . . , n. Then A1 , . . . , An are mutually disjoint 1.4.
PROVING THINGS FROM THE AXIOMS (each contains only one element which is in none
of the others), and A1 ? A2 ? · · · ? An = A; so by Axiom 3a, we have P(A) = P(a1 ) + P(a2 ) + · ·
· + P(an ). Corollary 1.2 If the sample space S is ?nite, say S = {a1 , . . . , an }, then P(a1 ) +
P(a2 ) + · · · + P(an ) = 1. For P(a1 ) + P(a2 ) + · · · + P(an ) = P(S ) by Proposition 1.1, and P(S )
= 1 by Axiom 2. Notice that once we have proved something, we can use it on the same basis as
an axiom to prove further facts. Now we see that, if all the n outcomes are equa.
This document provides an overview of probability concepts including:
- The three axioms of probability: probabilities are between 0 and 1, the probability of the sample space is 1, and the probability of the union of disjoint events equals the sum of the individual probabilities.
- Formulas for probability, conditional probability, independence, and complements.
- Discrete and continuous random variables and their properties including expected value and variance.
- Examples of probability mass functions for binomial and Poisson distributions.
Probabilistic information retrieval models & systemsSelman Bozkır
The document discusses probabilistic information retrieval and Bayesian approaches. It introduces concepts like conditional probability, Bayes' theorem, and the probability ranking principle. It explains how probabilistic models estimate the probability of relevance between a document and query by representing them as term sets and making probabilistic assumptions. The goal is to rank documents by the probability of relevance to present the most likely relevant documents first.
This document discusses key concepts in probability theory, including:
- Probability models random phenomena that may have deterministic or non-deterministic outcomes.
- The sample space defines all possible outcomes, and an event is any subset of outcomes.
- Probability is defined as the number of outcomes in an event divided by the total number of outcomes, if the sample space is finite and all outcomes are equally likely.
- Rules of probability include addition for mutually exclusive events and complement rules. Conditional probability adjusts probabilities based on additional information. Independence means events do not impact each other's probabilities.
Okay, let's solve this step-by-step:
1) Define the random variable:
X = Number of trips of 5 days or more per year
2) Write the probability distribution:
x P(x)
0 0.06
1 0.70
2 0.20
3 0.03
3) Calculate the mean using the formula:
Mean = Σx * P(x)
0 * 0.06 + 1 * 0.70 + 2 * 0.20 + 3 * 0.03 = 0.84
So the mean number of trips per year is 0.84.
The document discusses discrete probability concepts including sample spaces, events, axioms of probability, conditional probability, Bayes' theorem, random variables, probability distributions, expectation, and classical probability problems. It provides examples and explanations of key terms. The Monty Hall problem is used to demonstrate defining the sample space, event of interest, assigning probabilities, and computing the probability of winning by sticking or switching doors.
This document provides a probability cheatsheet compiled by William Chen and Joe Blitzstein with contributions from others. It is licensed under CC BY-NC-SA 4.0 and contains information on topics like counting rules, probability definitions, random variables, expectations, independence, and more. The cheatsheet is designed to summarize essential concepts in probability.
1. The document discusses basic concepts in probability and statistics, including sample spaces, events, probability distributions, and random variables.
2. Key concepts are explained such as independent and conditional probability, Bayes' theorem, and common probability distributions like the uniform and normal distributions.
3. Statistical analysis methods are introduced including how to estimate the mean and variance from samples from a distribution.
This document provides a review of key probability concepts for CS229 including:
- Random variables which map outcomes to real values
- Expectation and variance which are measures of the average and spread of a random variable
- Joint distributions and covariance which describe relationships between multiple random variables
- Conditioning, total probability, and Bayes' rule for relating probabilities of events
This document provides a probability cheatsheet compiled by William Chen and Joe Blitzstein with contributions from others. It is licensed under CC BY-NC-SA 4.0 and contains information on topics like counting rules, probability definitions, random variables, moments, and more. The cheatsheet is regularly updated with comments and suggestions submitted through a GitHub repository.
1 Probability Please read sections 3.1 – 3.3 in your .docxaryan532920
1
Probability
Please read sections 3.1 – 3.3 in your textbook
Def: An experiment is a process by which observations are generated.
Def: A variable is a quantity that is observed in the experiment.
Def: The sample space (S) for an experiment is the set of all possible outcomes.
Def: An event E is a subset of a sample space. It provides the collection of outcomes
that correspond to some classification.
Example:
Note: A sample space does not have to be finite.
Example: Pick any positive integer. The sample space is countably infinite.
A discrete sample space is one with a finite number of elements, { }1,2,3,4,5,6 or one that
has a countably infinite number of elements { }1,3,5,7,... .
A continuous sample space consists of elements forming a continuum. { }x / 2 x 5< <
2
A Venn diagram is used to show relationships between events.
A intersection B = (A ∩ B) = A and B
The outcomes in (A intersection B) belong to set A as well as to set B.
A union B = (A U B) = A alone or B alone or both
Union Formula
For any events A, B, P (A or B) = P (A) + P (B) – P (A intersection B) i.e.
P (A U B) = P (A) + P (B) – P (A ∩ B)
3
cA complement not A A ' A A = = = =
A complement consists of all outcomes outside of A.
Note: P (not A) = 1 – P (A)
Def: Two events are mutually exclusive (disjoint, incompatible) if they do not intersect,
i.e. if they do not occur at the same time. They have no outcomes in common.
When A and B are mutually exclusive, (A ∩ B) = null set = Ø, and P (A and B) = 0.
Thus, when A and B are mutually exclusive, P (A or B) = P (A) + P (B)
(This is exactly the same statement as rule 3 below)
Axioms of Probability
Def: A probability function p is a rule for calculating the probability of an event. The
function p satisfies 3 conditions:
1) 0 ≤ P (A) ≤1, for all events A in the sample space S
2) P (Sample Space S) = 1
3) If A, B, C are mutually exclusive events in the sample space S, then
P(A B C) P(A) P(B) P(C)∪ ∪ = + +
4
The Classical Probability Concept: If there are n equally likely possibilities, of which one
must occur and s are regarded as successes, then the probability of success is s
n
.
Example:
Frequency interpretation of Probability: The probability of an event E is the proportion of
times the event occurs during a long run of repeated experiments.
Example:
Def: A set function assigns a non-negative value to a set.
Ex: N (A) is a set function whose value is the number of elements in A.
Def: An additive set function f is a function for which f (A U B) = f (A) + f (B) when A and
B are mutually exclusive.
N (A) is an additive set function.
Ex: Toss 2 fair dice. Let A be the event that the sum on the two dice is 5. Let B be the
event that the sum on ...
- Probability theory studies possible outcomes of events and their likelihoods, expressed as a value from 0 to 1.
- Probability can be understood as the chance of an outcome, often expressed as a percentage between 0 and 100%.
- The analysis of data using probability models is called statistics.
The document discusses key concepts in probability theory including random variables, sample spaces, events, atomic events, laws of probability, conditional probabilities, independence, multivariate distributions, and Bayes' theorem. Random variables can be discrete or continuous. A sample space represents all possible outcomes and events are subsets of the sample space. The probability of the sample space is 1 and probabilities of events range from 0 to 1. Conditional probabilities and independence are discussed. Bayes' theorem provides a way to calculate conditional probabilities.
This document provides information about a probability and statistics course including the textbook, reference book, instructor, and an overview of key probability concepts like sample space, events, axioms of probability, joint probability, conditional probability, Bayes' theorem, statistical independence, and an example probability problem.
Slides for a lecture by Todd Davies on "Probability", prepared as background material for the Minds and Machines course (SYMSYS 1/PSYCH 35/LINGUIST 35/PHIL 99) at Stanford University. From a video recorded July 30, 2019, as part of a series of lectures funded by a Vice Provost for Teaching and Learning Innovation and Implementation Grant to the Symbolic Systems Program at Stanford, with post-production work by Eva Wallack. Topics include Basic Probability Theory, Conditional Probability, Independence, Philosophical Foundations, Subjective Probability Elicitation, and Heuristics and Biases in Human Probability Judgment.
LECTURE VIDEO: https://youtu.be/tqLluc36oD8
EDITED AND ENHANCED TRANSCRIPT: https://ssrn.com/abstract=3649241
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2. Sample Space
The possible outcomes of a random experiment
are called the basic outcomes, and the set of all
basic outcomes is called the sample space. The
symbol S will be used to denote the sample
space.
3. Sample Space
- An Example -
What is the sample space for a roll of a
single six-sided die?
S = [1, 2, 3, 4, 5, 6]
4. Mutually Exclusive
If the events A and B have no common basic outcomes,
they are mutually exclusive and their intersection A ∩ B
is said to be the empty set indicating that A ∩ B cannot
occur.
More generally, the K events E1, E2, . . . , EK are
said to be mutually exclusive if every pair of them is a
pair of mutually exclusive events.
5. Venn Diagrams
Venn Diagrams are drawings, usually using
geometric shapes, used to depict basic
concepts in set theory and the outcomes of
random experiments.
6. Intersection of Events A and B
S S
A A∩B B A B
(a) A∩B is the striped area (b) A and B are Mutually Exclusive
7. Collectively Exhaustive
Given the K events E1, E2, . . ., EK in the
sample space S. If E1 ∪ E2 ∪ . . . ∪EK = S,
these events are said to be collectively
exhaustive.
exhaustive
8. Complement
Let A be an event in the sample space S. The
set of basic outcomes of a random experiment
belonging to S but not to A is called the
complement of A and is denoted by A.
10. Unions, Intersections, and
Complements
A die is rolled. Let A be the event “Number rolled is even”
and B be the event “Number rolled is at least 4.” Then
A = [2, 4, 6] and B = [4, 5, 6]
A = [1, 3, 5] and B = [1, 2, 3]
A ∩ B = [4, 6]
A ∪ B = [2, 4, 5, 6]
A ∪ A = [1, 2, 3, 4, 5, 6] = S
11. Classical Probability
The classical definition of probability is the
proportion of times that an event will occur,
assuming that all outcomes in a sample space are
equally likely to occur. The probability of an
event is determined by counting the number of
outcomes in the sample space that satisfy the
event and dividing by the number of outcomes in
the sample space.
12. Classical Probability
The probability of an event A is
NA
P(A) =
N
where NA is the number of outcomes that satisfy the
condition of event A and N is the total number of outcomes
in the sample space. The important idea here is that one
can develop a probability from fundamental reasoning
about the process.
13. Combinations
The counting process can be generalized by
using the following equation to compare
the number of combinations of n things
taken k at a time.
n!
C =
n
k 0!= 1
k!(n − k )!
14. Relative Frequency
The relative frequency definition of probability is
the limit of the proportion of times that an
event A occurs in a large number of trials, n,
nA
P(A) =
n
where nA is the number of A outcomes and n is
the total number of trials or outcomes in the
population. The probability is the limit as n
becomes large.
15. Subjective Probability
The subjective definition of probability
expresses an individual’s degree of belief about
the chance that an event will occur. These
subjective probabilities are used in certain
management decision procedures.
16. Probability Postulates
Let S denote the sample space of a random experiment, Oi, the
basic outcomes, and A, an event. For each event A of the
sample space S, we assume that a number P(A) is defined
and we have the postulates
q If A is any event in the sample space S
0 ≤ P ( A) ≤ 1
q Let A be an event in S, and let Oi denote the basic outcomes.
Then
P ( A) = ∑ P (Oi )
A
where the notation implies that the summation extends over
all the basic outcomes in A.
3. P(S) = 1
17. Probability Rules
Let A be an event and A its complement.
The the complement rule is:
is
P ( A ) = 1 − P ( A)
18. Probability Rules
The Addition Rule of Probabilities:
Probabilities
Let A and B be two events. The probability of
their union is
P ( A ∪ B ) = P ( A) + P ( B ) − P ( A ∩ B )
19. Probability Rules
Venn Diagram for Addition Rule
P ( A ∪ B ) = P ( A) + P ( B ) − P ( A ∩ B )
P(A∪B)
A B
=
P(A) P(B) P(A∩B)
A B + A B - A B
20. Probability Rules
Conditional Probability:
Probability
Let A and B be two events. The conditional probability of
event A, given that event B has occurred, is denoted by the
symbol P(A|B) and is found to be:
P( A ∩ B)
P( A | B) =
P( B)
provided that P(B > 0).
21. Probability Rules
Conditional Probability:
Probability
Let A and B be two events. The conditional probability of
event B, given that event A has occurred, is denoted by the
symbol P(B|A) and is found to be:
P( A ∩ B)
P ( B | A) =
P ( A)
provided that P(A > 0).
22. Probability Rules
The Multiplication Rule of Probabilities:
Probabilities
Let A and B be two events. The probability of
their intersection can be derived from the
conditional probability as
P( A ∩ B) = P( A | B) P( B)
Also,
P ( A ∩ B ) = P ( B | A) P ( A)
23. Statistical Independence
Let A and B be two events. These events are said to be
statistically independent if and only if
P ( A ∩ B) = P( A) P ( B)
From the multiplication rule it also follows that
P(A | B) = P(A) (if P(B) > 0)
P(B | A) = P(B) (if P(A) > 0)
More generally, the events E1, E2, . . ., Ek are mutually
statistically independent if and only if
P(E1 ∩ E 2 ∩ ∩ E K ) = P(E1 ) P(E 2 ) P(E K )
25. Joint and Marginal Probabilities
In the context of bivariate probabilities, the
intersection probabilities P(Ai ∩ Bj) are called joint
probabilities. The probabilities for individual events
P(Ai) and P(Bj) are called marginal probabilities.
probabilities
Marginal probabilities are at the margin of a
bivariate table and can be computed by summing the
corresponding row or column.
26. Probabilities for the Television
Viewing and Income Example
Viewing High Middle Low Totals
Frequency Income Income
Income
Regular 0.04 0.13 0.04 0.21
Occasional 0.10 0.11 0.06 0.27
Never 0.13 0.17 0.22 0.52
Totals 0.27 0.41 0.32 1.00
28. Probability Rules
Rule for Determining the Independence of Attributes
Let A and B be a pair of attributes, each broken into
mutually exclusive and collectively exhaustive event
categories denoted by labels A1, A2, . . ., Ah and
B1, B2, . . ., Bk. If every Ai is statistically independent of
every event Bj, then the attributes A and B are
independent.
29. Bayes’ Theorem
Let A and B be two events. Then Bayes’ Theorem states
that:
P(A | B)P(B)
P( A | B) =
P(A)
and
P(B | A)P(A)
P( A | B) =
P(B)
30. Bayes’ Theorem
(Alternative Statement)
Let E1, E2, . . . , Ek be mutually exclusive and collectively
exhaustive events and let A be some other event. The
conditional probability of Ei given A can be expressed as
Bayes’ Theorem:
Theorem
P(A | E i )P(E i )
P(E i | A) =
P(A | E1 )P(E1 ) + P(A | E 2 )P(E 2 ) + + P(A | E K )P(E K )
31. Bayes’ Theorem
- Solution Steps -
1. Define the subset events from the
problem.
2. Define the probabilities for the events
defined in step 1.
3. Compute the complements of the
probabilities.
4. Apply Bayes’ theorem to compute the
probability for the problem solution.
33. Random Variables
A random variable is a variable that takes on
numerical values determined by the outcome
of a random experiment.
34. Discrete Random Variables
A random variable is discrete if it can
take on no more than a countable
number of values.
35. Discrete Random Variables
(Examples)
1. The number of defective items in a sample of twenty
items taken from a large shipment.
2. The number of customers arriving at a check-out
counter in an hour.
3. The number of errors detected in a corporation’s
accounts.
4. The number of claims on a medical insurance policy
in a particular year.
37. Continuous Random Variables
(Examples)
1. The income in a year for a family.
2. The amount of oil imported into the U.S. in a
particular month.
3. The change in the price of a share of IBM common
stock in a month.
4. The time that elapses between the installation of a new
computer and its failure.
5. The percentage of impurity in a batch of chemicals.
38. Discrete Probability Distributions
The probability distribution function (DPF), P(x),
of a discrete random variable expresses the
probability that X takes the value x, as a
function of x. That is
P ( x) = P ( X = x), for all values of x.
39. Discrete Probability Distributions
Graph the probability distribution function for
the roll of a single six-sided die.
P(x)
1/6
1 2 3 4 5 6 x
40. Required Properties of Probability
Distribution Functions of Discrete
Random Variables
Let X be a discrete random variable with
probability distribution function, P(x). Then
q P(x) ≥ 0 for any value of x
q The individual probabilities sum to 1; that is
∑ P( x) = 1
x
Where the notation indicates summation
over all possible values x.
41. Cumulative Probability Function
The cumulative probability function, F(x0), of a
random variable X expresses the probability
that X does not exceed the value x0, as a
function of x0. That is
F ( x0 ) = P ( X ≤ x0 )
Where the function is evaluated at all values x0
42. Derived Relationship Between Probability
Function and Cumulative Probability
Function
Let X be a random variable with probability function
P(x) and cumulative probability function F(x0). Then it
can be shown that
F ( x0 ) = ∑ P( X )
x ≤ x0
Where the notation implies that summation is over all
possible values x that are less than or equal to x0.
43. Derived Properties of Cumulative
Probability Functions for Discrete
Random Variables
Let X be a discrete random variable with a
cumulative probability function, F(x0).
Then we can show that
q 0 ≥ F(x0) ≥ 1 for every number x0
q If x0 and x1 are two numbers with x0 < x1,
then F(x0) ≤ F(x1)
44. Expected Value
The expected value, E(X), of a discrete random
variable X is defined
E ( X ) = ∑ xP( x)
x
Where the notation indicates that summation extends
over all possible values x.
The expected value of a random variable is called its
mean and is denoted µx.
45. Expected Value: Functions of
Random Variables
Let X be a discrete random variable with
probability function P(x) and let g(X) be some
function of X. Then the expected value, E[g(X)],
of that function is defined as
E[ g ( X )] = ∑ g ( x) P ( x)
x
46. Variance and Standard Deviation
Let X be a discrete random variable. The expectation
of the squared discrepancies about the mean, (X - µ)2,
is called the variance, denoted σ2x and is given by
variance
σ x = E ( X − µ x ) 2 = ∑ ( x − µ x ) 2 P( x)
2
x
The standard deviation, σx , is the positive square root
deviation
of the variance.
47. Variance
(Alternative Formula)
The variance of a discrete random variable X can be
Expressed as
σ = E( X ) − µx
2 2 2
x
= ∑ x P( x) − µ x
2 2
x
48. Expected Value and Variance for
Discrete Random Variable Using
Microsoft Excel
Sales P(x) Mean Variance
0 0.15 0 0.570375
1 0.3 0.3 0.27075
2 0.2 0.4 0.0005
3 0.2 0.6 0.2205
4 0.1 0.4 0.42025
5 0.05 0.25 0.465125
1.95 1.9475
Expected Value = 1.95 Variance = 1.9475
49. Summary of Properties for Linear
Function of a Random Variable
Let X be a random variable with mean µx , and variance σ2x
; and let a and b be any constant fixed numbers. Define the
random variable Y = a + bX. Then, the mean and variance
of Y are
µY = E (a + bX ) = a + bµ X
and
σ 2
Y = Var (a + bX ) = b σ
2 2
X
so that the standard deviation of Y is
σY = bσ X
50. Summary Results for the Mean and
Variance of Special Linear Functions
q Let b = 0 in the linear function, W = a + bX. Then W = a
(for any constant a).
E (a) = a and Var (a ) = 0
If a random variable always takes the value a, it will have a
mean a and a variance 0.
q Let a = 0 in the linear function, W = a + bX. Then W =
bX.
E (bX ) = bµ X and Var (a ) = b 2σ X
2
51. Mean and Variance of Z
Let a = -µX/σX and b = 1/ σX in the linear function Z = a
+ bX. Then,
X − µX
Z = a + bX =
σX
so that
X − µX µX 1
E
σ =−
+ µX = 0
X σX σX
and
X − µX 1 2
Var
σ = 2 σ X =1
σ
X X
52. Bernoulli Distribution
A Bernoulli distribution arises from a random experiment
which can give rise to just two possible outcomes. These
outcomes are usually labeled as either “success” or
“failure.” If π denotes the probability of a success and the
probability of a failure is (1 - π ), the the Bernoulli
probability function is
P (0) = (1 − π ) and P (1) = π
53. Mean and Variance of a Bernoulli
Random Variable
The mean is:
µ X = E ( X ) = ∑ xP( x) = (0)(1 − π ) + (1)π = π
X
And the variance is:
σ = E[( X − µ X ) ] = ∑ ( x − µ X ) P( x)
2
X
2 2
X
= (0 − π ) 2 (1 − π ) + (1 − π ) 2 π = π (1 − π )
54. Sequences of x Successes in n
Trials
The number of sequences with x successes in n independent
trials is:
n!
C =
n
x
x!(n − x)!
Where n! = n x (x – 1) x (n – 2) x . . . x 1 and 0! = 1.
n
These C x sequences are mutually exclusive,
since no two of them can occur at the same time.
55. Binomial Distribution
Suppose that a random experiment can result in two possible mutually
exclusive and collectively exhaustive outcomes, “success” and “failure,”
and that π is the probability of a success resulting in a single trial. If n
independent trials are carried out, the distribution of the resulting
number of successes “x” is called the binomial distribution. Its
distribution
probability distribution function for the binomial random variable X =
x is:
P(x successes in n independent trials)=
n! ( n− x )
P( x) = π (1 − π )
x
x!(n − x)!
for x = 0, 1, 2 . . . , n
56. Mean and Variance of a Binomial
Probability Distribution
Let X be the number of successes in n independent trials,
each with probability of success π. The x follows a
binomial distribution with mean,
mean
µ X = E ( X ) = nπ
and variance,
variance
σ = E[( X − µ ) ] = nπ (1 − π )
2
X
2
57. Binomial Probabilities
- An Example –
An insurance broker, has five contracts, and he believes
that for each contract, the probability of making a sale is
0.40.
What is the probability that he makes at most one sale?
P(at most one sale) = P(X ≤ 1) = P(X = 0) + P(X = 1)
= 0.078 + 0.259 = 0.337
5!
P(no sales) = P(0) = (0.4) 0 (0.6) 5 = 0.078
0!5!
5!
P(1 sale) = P(1) = (0.4)1 (0.6) 4 = 0.259
1!4!
59. Poisson Probability Distribution
Assume that an interval is divided into a very large number of
subintervals so that the probability of the occurrence of an
event in any subinterval is very small. The assumptions of
a Poisson probability distribution are:
2) The probability of an occurrence of an event is constant for
all subintervals.
3) There can be no more than one occurrence in each
subinterval.
4) Occurrences are independent; that is, the number of
occurrences in any non-overlapping intervals in
independent of one another.
60. Poisson Probability Distribution
The random variable X is said to follow the Poisson
probability distribution if it has the probability function:
e − λ λx
P( x) = , for x = 0, 1,2,...
x!
where
P(x) = the probability of x successes over a given period of
time or space, given λ
λ = the expected number of successes per time or space
unit; λ > 0
e = 2.71828 (the base for natural logarithms)
The mean and variance of the Poisson probability distribution are:
are
µ x = E ( X ) = λ and σ x2 = E[( X − µ ) 2 ] = λ
62. Poisson Approximation to the
Binomial Distribution
Let X be the number of successes resulting from n independent
trials, each with a probability of success, π. The distribution of the
number of successes X is binomial, with mean nπ. If the number of
trials n is large and nπ is of only moderate size (preferably nπ ≤ 7),
this distribution can be approximated by the Poisson distribution
with λ = nπ. The probability function of the approximating
distribution is then:
e − nπ (nπ ) x
P( x) = , for x = 0, 1,2,...
x!
63. Covariance
Let X be a random variable with mean µ X , and let Y be a
random variable with mean, µ Y . The expected value of (X -
µ X )(Y - µ Y ) is called the covariance between X and Y,
denoted Cov(X, Y).
For discrete random variables
Cov ( X , Y ) = E[( X − µ X )(Y − µY )] = ∑∑ ( x − µ x )( y − µ y ) P ( x, y )
x y
An equivalent expression is
Cov ( X , Y ) = E ( XY ) − µ x µ y = ∑∑ xyP( x, y ) − µ x µ y
x y
64. Correlation
Let X and Y be jointly distributed random variables.
The correlation between X and Y is:
Cov ( X , Y )
ρ = Corr ( X , Y ) =
σ XσY
65. Covariance and Statistical
Independence
If two random variables are statistically
independent, the covariance between them is 0.
independent
However, the converse is not necessarily true.
66. Portfolio Analysis
The random variable X is the price for stock A and the
random variable Y is the price for stock B. The market
value, W, for the portfolio is given by the linear function,
W = aX + bY
Where, a, is the number of shares of stock A and, b, is the
number of shares of stock B.
67. Portfolio Analysis
The mean value for W is,
µW = E[W ] = E[aX + bY ]
= aµ X + bµY
The variance for W is,
σ = a σ + b σ + 2abCov ( X , Y )
2
W
2 2
X
2 2
Y
or using the correlation,
σ = a σ + b σ + 2abCorr ( X , Y )σ X σ Y
2
W
2 2
X
2 2
Y
70. Cumulative Distribution Function
The cumulative distribution function, F(x), for a
function
continuous random variable X expresses the
probability that X does not exceed the value of x, as
a function of x
F ( x) = P( X ≤ x)
72. Cumulative Distribution Function
Let X be a continuous random variable with a
cumulative distribution function F(x), and let a and
b be two possible values of X, with a < b. The
probability that X lies between a and b is
P(a < X < b) = F (b) − F (a )
73. Probability Density Function
Let X be a continuous random variable, and let x be any number lying in
the range of values this random variable can take. The probability density
function, f(x), of the random variable is a function with the following
function
properties:
q f(x) > 0 for all values of x
q The area under the probability density function f(x) over all values of the
random variable X is equal to 1.0
q Suppose this density function is graphed. Let a and b be two possible
values of the random variable X, with a<b. Then the probability that X lies
between a and b is the area under the density function between these points.
q The cumulative density function F(x0) is the area under the probability
density function f(x) up to x0
x0
f ( x0 ) = ∫ f ( x)dx
xm
74. Shaded Area is the Probability That
X is Between a and b
0 a b x
76. Areas Under Continuous Probability
Density Functions
Let X be a continuous random variable with the
probability density function f(x) and cumulative
distribution F(x). Then the following properties
hold:
q The total area under the curve f(x) = 1.
q The area under the curve f(x) to the left of x0 is
F(x0), where x0 is any value that the random
variable can take.
77. Properties of the Probability Density
Function
f(x)
Comments
1
Total area under
the uniform
probability density
function is 1.
0
0 x0 1 x
78. Properties of the Probability Density
Function
Comments
f(x)
Area under the uniform
probability density
function to the left of
1
x0 is F(x0), which is
equal to x0 for this
uniform distribution
because f(x)=1.
0
0 x0 1 x
79. Rationale for Expectations of
Continuous Random Variables
Suppose that a random experiment leads to an
outcome that can be represented by a continuous
random variable. If N independent replications of
this experiment are carried out, then the expected
value of the random variable is the average of the
values taken, as the number of replications becomes
infinitely large. The expected value of a random
variable is denoted by E(X).
80. Rationale for Expectations of
Continuous Random Variables
(continued)
Similarly, if g(x) is any function of the random
variable, X, then the expected value of this function is
the average value taken by the function over repeated
independent trials, as the number of trials becomes
infinitely large. This expectation is denoted E[g(X)].
By using calculus we can define expected values for
continuous random variables similarly to that used for
discrete random variables.
E[ g ( x)] = ∫ g ( x) f ( x)dx
x
81. Mean, Variance, and Standard
Deviation
Let X be a continuous random variable. There are two important expected values
that are used routinely to define continuous probability distributions.
q The mean of X, denoted by µX, is defined as the expected value of X.
X
µ X = E( X )
q The variance of X, denoted by σX2, is defined as the expectation of the
X
squared deviation, (X - µX)2, of a random variable from its mean
σ = E[( X − µ X ) ]
2
X
2
Or an alternative expression can be derived
σ X = E( X 2 ) − µ X
2 2
q The standard deviation of X, σX, is the square root of the variance.
X
82. Linear Functions of Variables
Let X be a continuous random variable with mean µ X and
variance σ X2, and let a and b any constant fixed numbers.
Define the random variable W as
W = a + bX
Then the mean and variance of W are
µW = E (a + bX ) = a + bµ X
and
σ = Var (a + bX ) = b σ
2
W
2 2
X
and the standard deviation of W is
σW = bσ X
83. Linear Functions of Variable
(continued)
An important special case of the previous results is the
standardized random variable
X − µX
Z=
σX
which has a mean 0 and variance 1.
84. Reasons for Using the Normal
Distribution
1. The normal distribution closely approximates the
probability distributions of a wide range of random
variables.
2. Distributions of sample means approach a normal
distribution given a “large” sample size.
3. Computations of probabilities are direct and
elegant.
4. The normal probability distribution has led to good
business decisions for a number of applications.
86. Probability Density Function of
the Normal Distribution
The probability density function for a normally
distributed random variable X is
1 − ( x − µ ) 2 / 2σ 2
f ( x) = e for - ∞ < x < ∞
2πσ 2
Where µ and σ 2 are any number such that -∞ < µ < ∞
and -∞ < σ 2 < ∞ and where e and π are physical
constants, e = 2.71828. . . and π = 3.14159. . .
87. Properties of the Normal
Distribution
Suppose that the random variable X follows a normal distribution with
parameters µ and σ2. Then the following properties hold:
q The mean of the random variable is µ,
E( X ) = µ
q The variance of the random variable is σ2,
E[( X − µ X ) 2 ] = σ 2
q The shape of the probability density function is a symmetric bell-
shaped curve centered on the mean µ as shown in Figure 6.8.
vii. By knowing the mean and variance we can define the normal
distribution by using the notation
X ~ N (µ ,σ ) 2
88. Effects of µ on the Probability Density
Function of a Normal Random Variable
0.4
0.3 Mean = 5 Mean = 6
0.2
0.1
0.0
1.5 2.5 3.5 4.5 5.5 6.5 7.5 8.5 x
89. Effects of σ2 on the Probability Density
Function of a Normal Random Variable
0.4 Variance = 0.0625
0.3
0.2
Variance = 1
0.1
0.0
1.5 2.5 3.5 4.5 5.5 6.5 7.5 8.5
x
90. Cumulative Distribution Function
of the Normal Distribution
Suppose that X is a normal random variable with mean
µ and variance σ 2 ; that is X~N(µ, σ 2). Then the
cumulative distribution function is
F ( x0 ) = P ( X ≤ x0 )
This is the area under the normal probability density
function to the left of x0, as illustrated in Figure 6.10. As
for any proper density function, the total area under the
curve is 1; that is F(∞) = 1.
91. Shaded Area is the Probability that X
does not Exceed x0 for a Normal
Random Variable
f(x)
x0 x
92. Range Probabilities for Normal
Random Variables
Let X be a normal random variable with cumulative
distribution function F(x), and let a and b be two
possible values of X, with a < b. Then
P (a < X < b) = F (b) − F (a )
The probability is the area under the corresponding
probability density function between a and b.
94. The Standard Normal Distribution
Let Z be a normal random variable with mean 0 and
variance 1; that is
Z ~ N (0,1)
We say that Z follows the standard normal distribution.
Denote the cumulative distribution function as F(z), and a
and b as two numbers with a < b, then
P (a < Z < b) = F (b) − F (a)
96. Finding Range Probabilities for Normally
Distributed Random Variables
Let X be a normally distributed random variable with mean µ
and variance σ 2. Then the random variable Z = (X - µ)/σ has a
standard normal distribution: Z ~ N(0, 1)
It follows that if a and b are any numbers with a < b, then
a−µ b−µ
P ( a < X < b) = P <Z<
σ σ
b−µ a−µ
= F − F
σ σ
where Z is the standard normal random variable and F(z) denotes
its cumulative distribution function.
97. Computing Normal Probabilities
A very large group of students obtains test scores that are
normally distributed with mean 60 and standard deviation 15.
What proportion of the students obtained scores between 85
and 95?
85 − 60 95 − 60
P (85 < X < 95) = P <Z<
15 15
= P (1.67 < Z < 2.33)
= F (2.33) − F (1.67)
= 0.9901 − 0.9525 = 0.0376
That is, 3.76% of the students obtained scores in the range 85 to 95.
98. Approximating Binomial Probabilities
Using the Normal Distribution
Let X be the number of successes from n independent Bernoulli
trials, each with probability of success π. The number of successes,
X, is a Binomial random variable and if nπ(1 - π) > 9 a good
approximation is
a − nπ b − nπ
P ( a < X < b) = P ≤Z≤
nπ (1 − π ) nπ (1 − π )
Or if 5 < nπ(1 - π) < 9 we can use the continuity correction factor to
obtain
a − 0.5 − nπ b + 0.5 − nπ
P ( a ≤ X ≤ b) = P ≤Z≤
nπ (1 − π ) nπ (1 − π )
where Z is a standard normal variable.
99. Covariance
Let X and Y be a pair of continuous random variables,
with respective means µ x and µ y. The expected value of (x
- µ x)(Y - µ y) is called the covariance between X and Y.
That is
Cov( X , Y ) = E[( X − µ x )(Y − µ y )]
An alternative but equivalent expression can be derived as
Cov( X , Y ) = E ( XY ) − µ x µ y
If the random variables X and Y are independent, then the
covariance between them is 0. However, the converse is
not true.
100. Correlation
Let X and Y be jointly distributed random variables. The
correlation between X and Y is
Cov( X , Y )
ρ = Corr ( X , Y ) =
σ XσY
101. Sums of Random Variables
Let X1, X2, . . .Xk be k random variables with means µ1, µ2,. . .
µk and variances σ12, σ22,. . ., σk2. The following properties
hold:
ii. The mean of their sum is the sum of their means; that is
E ( X 1 + X 2 + + X k ) = µ1 + µ 2 + + µ k
iv. If the covariance between every pair of these random
variables is 0, then the variance of their sum is the sum of
their variances; that is
Var ( X 1 + X 2 + + X k ) = σ 12 + σ 2 + + σ k2
2
However, if the covariances between pairs of random
variables are not 0, the variance of their sum is
K −1 K
Var ( X 1 + X 2 + + X k ) = σ 12 + σ 2 + + σ k2 + 2∑
2
∑ Cov( X , X
i j )
i =1 j =i +1
102. Differences Between a Pair of
Random Variables
Let X and Y be a pair of random variables with means µX and µY and
variances σX2 and σY2. The following properties hold:
ii. The mean of their difference is the difference of their means; that is
E ( X − Y ) = µ X − µY
iv. If the covariance between X and Y is 0, then the variance of their
difference is
Var ( X − Y ) = σ X + σ Y
2 2
vi. If the covariance between X and Y is not 0, then the variance of their
difference is
Var ( X − Y ) = σ X + σ Y − 2Cov ( X , Y )
2 2
103. Linear Combinations of Random
Variables
The linear combination of two random variables, X and Y, is
W = aX + bY
Where a and b are constant numbers.
The mean for W is,
µW = E[W ] = E[aX + bY ] = aµ X + bµY
The variance for W is,
σ W = a 2σ X + b 2σ Y + 2abCov ( X , Y )
2 2 2
Or using the correlation,
σ W = a 2σ X + b 2σ Y + 2abCorr ( X , Y )σ X σ Y
2 2 2
If both X and Y are joint normally distributed random variables
then the resulting random variable, W, is also normally distributed
with mean and variance derived above.