The document discusses conditional probability mass functions (PMFs). A conditional PMF characterizes the probability distribution of a discrete random variable X given that some conditioning event B has occurred. It is defined as PX|B(x) = P(X = x | B). Like a regular PMF, a conditional PMF is a function between 0 and 1 whose values sum to 1. Conditional PMFs are useful for compound experiments where the second part depends on the first. They allow probabilities to be computed using the law of total probability and conditional expected values.
This document provides a concise probability cheatsheet compiled by William Chen and others. It covers key probability concepts like counting rules, sampling tables, definitions of probability, independence, unions and intersections, joint/marginal/conditional probabilities, Bayes' rule, random variables and their distributions, expected value, variance, indicators, moment generating functions, and independence of random variables. The cheatsheet is licensed under CC BY-NC-SA 4.0 and the last updated date is March 20, 2015.
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.
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. The document covers probability axioms and rules including the additive rule, conditional probability, independence, and Bayes' rule. It also defines discrete and continuous random variables and their probability distributions.
2. Important discrete distributions discussed include the Bernoulli distribution for a binary outcome experiment and the binomial distribution for repeated Bernoulli trials.
3. Techniques for counting permutations, combinations, and sequences of events are presented to handle probability problems involving counting.
The document discusses Bayes' rule and entropy in data mining. It provides step-by-step derivations of Bayes' rule from definitions of conditional probability and the chain rule. It then gives examples of calculating entropy for variables with different probability distributions, noting that maximum entropy occurs with a uniform distribution where all outcomes are equally likely, while minimum entropy occurs when the probability of one outcome is 1.
this materials is useful for the students who studying masters level in elect...BhojRajAdhikari5
This document discusses concepts related to continuous and discrete random variables including:
- Defining continuous random variables based on their cumulative distribution function (CDF) being continuous.
- Introducing the probability density function (PDF) as the derivative of the CDF, which indicates the probability of a continuous random variable being near a given value.
- Defining joint and conditional probability distribution functions for multiple random variables.
- Discussing statistical independence and functions of random variables.
- Introducing statistical averages like the expected value, variance, and standard deviation of random variables. Formulas for calculating these are provided.
Accounting for uncertainty is a crucial component in decision making (e.g., classification) because of ambiguity in our measurements.
Probability theory is the proper mechanism for accounting for uncertainty.
On Local Integrable Solutions of Abstract Volterra Integral EquationsIOSR Journals
This document discusses local integrable solutions of abstract Volterra integral equations. It begins with an introduction that provides context on integral equations and previous work studying locally integrable solutions. It then outlines the mathematical framework and definitions used, including topological spaces, measure spaces, function spaces, and conditions on operators. The main results section proves two theorems: 1) an operator maps locally integrable functions to continuous functions under certain conditions, and 2) an operator maps a function space to itself continuously under additional conditions. An example is also mentioned to illustrate the results.
This document provides a concise probability cheatsheet compiled by William Chen and others. It covers key probability concepts like counting rules, sampling tables, definitions of probability, independence, unions and intersections, joint/marginal/conditional probabilities, Bayes' rule, random variables and their distributions, expected value, variance, indicators, moment generating functions, and independence of random variables. The cheatsheet is licensed under CC BY-NC-SA 4.0 and the last updated date is March 20, 2015.
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.
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. The document covers probability axioms and rules including the additive rule, conditional probability, independence, and Bayes' rule. It also defines discrete and continuous random variables and their probability distributions.
2. Important discrete distributions discussed include the Bernoulli distribution for a binary outcome experiment and the binomial distribution for repeated Bernoulli trials.
3. Techniques for counting permutations, combinations, and sequences of events are presented to handle probability problems involving counting.
The document discusses Bayes' rule and entropy in data mining. It provides step-by-step derivations of Bayes' rule from definitions of conditional probability and the chain rule. It then gives examples of calculating entropy for variables with different probability distributions, noting that maximum entropy occurs with a uniform distribution where all outcomes are equally likely, while minimum entropy occurs when the probability of one outcome is 1.
this materials is useful for the students who studying masters level in elect...BhojRajAdhikari5
This document discusses concepts related to continuous and discrete random variables including:
- Defining continuous random variables based on their cumulative distribution function (CDF) being continuous.
- Introducing the probability density function (PDF) as the derivative of the CDF, which indicates the probability of a continuous random variable being near a given value.
- Defining joint and conditional probability distribution functions for multiple random variables.
- Discussing statistical independence and functions of random variables.
- Introducing statistical averages like the expected value, variance, and standard deviation of random variables. Formulas for calculating these are provided.
Accounting for uncertainty is a crucial component in decision making (e.g., classification) because of ambiguity in our measurements.
Probability theory is the proper mechanism for accounting for uncertainty.
On Local Integrable Solutions of Abstract Volterra Integral EquationsIOSR Journals
This document discusses local integrable solutions of abstract Volterra integral equations. It begins with an introduction that provides context on integral equations and previous work studying locally integrable solutions. It then outlines the mathematical framework and definitions used, including topological spaces, measure spaces, function spaces, and conditions on operators. The main results section proves two theorems: 1) an operator maps locally integrable functions to continuous functions under certain conditions, and 2) an operator maps a function space to itself continuously under additional conditions. An example is also mentioned to illustrate the results.
1) Probability is defined as a set function that satisfies three axioms: non-negativity, the probability of the sample space is 1, and countable additivity.
2) Conditional probability is the probability of an event B given that event A has occurred, defined as P(B|A)=P(A∩B)/P(A). Events A and B are independent if P(B|A)=P(B) and P(A|B)=P(A).
3) Bayes' theorem gives the probability of an event A given that event B has occurred as P(A|B)=P(A)P(B|A)/P(B).
This document contains permissions and copyright information for Chapter 2 of the Handbook of Applied Cryptography. It grants permission to retrieve, print, and store a single copy of this chapter for personal use, but does not extend permission to bind multiple chapters, photocopy, produce additional copies, or make electronic copies available without prior written permission. Except as specifically permitted, the standard copyright from CRC Press applies and prohibits reproducing or transmitting the book or any part in any form without prior written permission.
This document discusses moment-generating functions and their properties and applications in probability theory. It defines the moment-generating function for both discrete and continuous random variables. Some key properties of moment-generating functions are that each probability distribution has a unique moment-generating function, and they can be used to find the distribution of sums of random variables. The document also describes several common probability distributions and derives their corresponding moment-generating functions.
Moment-Generating Functions and Reproductive Properties of DistributionsIJSRED
The document discusses moment-generating functions and their applications in probability theory. It defines the moment-generating function for both discrete and continuous random variables. Some key properties and examples of moment-generating functions for common distributions like binomial, Poisson, exponential, normal and gamma are provided. Reproductive properties of distributions are also described briefly.
The document discusses cumulative distribution functions (CDFs) and probability density functions (PDFs) for continuous random variables. It provides definitions and properties of CDFs and PDFs. For CDFs, it describes how they give the probability that a random variable is less than or equal to a value. For PDFs, it explains how they provide the probability of a random variable taking on a particular value. The document also gives examples of CDFs and PDFs for exponential and uniform random variables.
The document discusses different types of random variables including discrete, continuous, Bernoulli, binomial, geometric, Poisson, and uniform random variables. It provides the definitions and probability mass/density functions for each type. Examples are also given to illustrate concepts such as calculating probabilities for different random variables.
1. A random variable (RV) is a function that maps outcomes of a random phenomenon to numerical values. For a function X to be an RV, the set P(x) = {ω: X(ω) ≤ x} must be an event with a well-defined probability P for any x.
2. Conditional probability P(A|B) is the probability of event A occurring given that event B has occurred. It is defined as P(A ∩ B)/P(B).
3. The law of total probability states that for mutually exclusive and exhaustive events B1, B2, ..., the probability of event A is the sum of the probabilities of A given each Bi multiplied by
An Algorithm For The Combined Distribution And Assignment ProblemAndrew Parish
This document presents an algorithm for solving the combined distribution and assignment problem using generalized Benders' decomposition. The algorithm formulates the problem as a modified distribution problem with a minimax objective function instead of a linear one. It solves this master problem using the Newton-Kantorovich method for nonlinear concave programming problems with linear constraints. The algorithm iterates between solving the assignment problem given a distribution and solving the modified distribution problem subject to optimality constraints from the assignment problem. When the solution converges, it provides the optimal traffic flows for both distribution and assignment.
Chapter 1 - Probability Distributions.pdfMayurMandhub
This document outlines the key concepts covered in the Quantitative Finance 2 course. It discusses probability distributions, including defining random variables and their properties. It covers discrete and continuous probability distributions, and the use of probability density functions and cumulative distribution functions. The document also summarizes important probability rules like addition rule, multiplication rule, conditional probability, total probability rule, and Bayes' theorem. Sample problems and examples are provided to illustrate concepts like independent and dependent events, and how to calculate conditional and posterior probabilities.
The document discusses random processes and probability theory concepts relevant to communication systems. It defines key terms like random variables, sample space, events, probability, and distributions. It describes different types of random variables like discrete and continuous, and their probability mass functions and density functions. It also discusses statistical measures like mean, variance, and covariance that are used to characterize random signals and compare signals. Specific random variables discussed include binomial and uniform. The document provides foundations for analyzing random signals in communication systems.
This document provides an introduction to the author's work on developing a unified treatment of elliptic, hyperbolic, and partly elliptic-hyperbolic differential equations using symmetric positive linear operators. The author aims to show that these types of equations can be treated within a single framework by formulating the equations as systems of first-order equations and imposing admissible boundary conditions. Specifically:
1) The author introduces symmetric positive linear differential operators that satisfy certain algebraic properties and uses these to formulate differential equations as systems of first-order equations.
2) Admissible boundary conditions are defined that depend only on the nature of the operator coefficients on the boundary.
3) It is shown that under these conditions, the boundary value problems
Conditional mixture model and its application for regression modelLoc Nguyen
Expectation maximization (EM) algorithm is a powerful mathematical tool for estimating statistical parameter when data sample contains hidden part and observed part. EM is applied to learn finite mixture model in which the whole distribution of observed variable is average sum of partial distributions. Coverage ratio of every partial distribution is specified by the probability of hidden variable. An application of mixture model is soft clustering in which cluster is modeled by hidden variable whereas each data point can be assigned to more than one cluster and degree of such assignment is represented by the probability of hidden variable. However, such probability in traditional mixture model is simplified as a parameter, which can cause loss of valuable information. Therefore, in this research I propose a so-called conditional mixture model (CMM) in which the probability of hidden variable is modeled as a full probabilistic density function (PDF) that owns individual parameter. CMM aims to extend mixture model. I also propose an application of CMM which is called adaptive regression model (ARM). Traditional regression model is effective when data sample is scattered equally. If data points are grouped into clusters, regression model tries to learn a unified regression function which goes through all data points. Obviously, such unified function is not effective to evaluate response variable based on grouped data points. The concept “adaptive” of ARM means that ARM solves the ineffectiveness problem by selecting the best cluster of data points firstly and then evaluating response variable within such best cluster. In orther words, ARM reduces estimation space of regression model so as to gain high accuracy in calculation.
Keywords: expectation maximization (EM) algorithm, finite mixture model, conditional mixture model, regression model, adaptive regression model (ARM).
Noise is unwanted sound considered unpleasant, loud, or disruptive to hearing. From a physics standpoint, there is no distinction between noise and desired sound, as both are vibrations through a medium, such as air or water. The difference arises when the brain receives and perceives a sound.
Error Estimates for Multi-Penalty Regularization under General Source Conditioncsandit
In learning theory, the convergence issues of the regression problem are investigated with
the least square Tikhonov regularization schemes in both the RKHS-norm and the L 2
-norm.
We consider the multi-penalized least square regularization scheme under the general source
condition with the polynomial decay of the eigenvalues of the integral operator. One of the
motivation for this work is to discuss the convergence issues for widely considered manifold
regularization scheme. The optimal convergence rates of multi-penalty regularizer is achieved
in the interpolation norm using the concept of effective dimension. Further we also propose
the penalty balancing principle based on augmented Tikhonov regularization for the choice of
regularization parameters. The superiority of multi-penalty regularization over single-penalty
regularization is shown using the academic example and moon data set.
a) Use Newton’s Polynomials for Evenly Spaced data to derive the O(h.pdfpetercoiffeur18
a) Use Newton’s Polynomials for Evenly Spaced data to derive the O(h4) accurate Second
Centered Difference approximation of the 1st derivative at nx. Start with a polynomial fit to
points at n-2x , n-1x, nx , n+1x and n+2x .
b) Use Newton’s Polynomials for Evenly Spaced data to derive the O(h4) accurate Second
Centered Difference approximation of the 2nd derivative at nx . Remember, to keep the same
O(h4) accuracy, while taking one more derivative than in Part a, we need to add a point to the
polynomial we used in part a.t,s01530456075y,km0356488107120
Solution
An interpolation assignment generally entails a given set of information points: in which the
values yi can,
xi x0 x1 ... xn
f(xi) y0 y1 ... yn
for instance, be the result of a few bodily measurement or they can come from a long
numerical calculation. hence we know the fee of the underlying characteristic f(x) at the set
of points xi, and we want to discover an analytic expression for f .
In interpolation, the assignment is to estimate f(x) for arbitrary x that lies among the smallest
and the most important xi
. If x is out of doors the variety of the xi’s, then the task is called extrapolation,
which is substantially greater unsafe.
with the aid of far the maximum not unusual useful paperwork utilized in interpolation are the
polynomials.
different picks encompass, as an instance, trigonometric functions and spline features
(mentioned
later during this direction).
Examples of different sorts of interpolation responsibilities include:
1. Having the set of n + 1 information factors xi
, yi, we want to understand the fee of y in the
complete c program languageperiod x = [x0, xn]; i.e. we need to find a simple formulation
which reproduces
the given points exactly.
2. If the set of statistics factors contain errors (e.g. if they are measured values), then we
ask for a components that represents the records, and if feasible, filters out the errors.
3. A feature f may be given within the shape of a pc system which is high priced
to assess. In this case, we want to find a characteristic g which offers a very good
approximation of f and is simpler to assess.
2 Polynomial interpolation
2.1 Interpolating polynomial
Given a fixed of n + 1 records points xi
, yi, we need to discover a polynomial curve that passes
via all the factors. as a consequence, we search for a non-stop curve which takes at the values yi
for every of the n+1 wonderful xi’s.
A polynomial p for which p(xi) = yi whilst zero i n is stated to interpolate the given set of
records points. The factors xi are known as nodes.
The trivial case is n = zero. right here a steady function p(x) = y0 solves the hassle.
The only case is n = 1. In this situation, the polynomial p is a directly line described via
p(x) =
xx1
x0 x1
y0 +
xx0
x1 x0
y1
= y0 +
y1 y0
x1 x0
(xx0)
here p is used for linear interpolation.
As we will see, the interpolating polynomial may be written in an expansion of paperwork,
among
these are the Newton shape and the Lag.
This document discusses using the Newton-Raphson iterative method to solve chemical equilibrium problems. It begins by introducing fixed point theory and the Newton-Raphson method for solving nonlinear equations. It then describes applying this method to determine the O reactant ratio that produces an adiabatic equilibrium temperature in the chemical reaction of partial methane oxidation. Specifically, it develops a system of seven nonlinear equations and uses the Newton-Raphson method to iteratively solve for the fixed point and desired chemical equilibrium conditions.
On Analytic Review of Hahn–Banach Extension Results with Some GeneralizationsBRNSS Publication Hub
The useful Hahn–Banach theorem in functional analysis has significantly been in use for many years ago. At this point in time, we discover that its domain and range of existence can be extended point wisely so as to secure a wider range of extendibility. In achieving this, we initially reviewed the existing traditional Hahn–Banach extension theorem, before we carefully and successfully used it to generate the finite extension form as in main results of section three.
This document provides an overview and proofs of several theorems related to the Hahn-Banach theorem. It begins with an introduction to linear functionals and the Hahn-Banach theorem. It then presents two main theorems - the Hahn-Banach theorem and the topological Hahn-Banach theorem. The document provides proofs of these theorems and several related theorems using the Hahn-Banach extension lemma. It also discusses consequences of the Hahn-Banach extension form and provides proofs of the theorems using the lemma.
This document provides an overview and definitions related to functional analysis and Banach spaces. It discusses:
1) The definition of a Banach space as a complete normed linear space where every Cauchy sequence converges.
2) Examples of Banach spaces including l^p spaces, C(X) for compact X, C_b(X) for any topological space X, and C^k([a,b]).
3) Measure spaces and the definition of measurable functions on a measure space. It notes the closure properties of measurable functions under scalar multiplication and (sometimes) addition.
1) Probability is defined as a set function that satisfies three axioms: non-negativity, the probability of the sample space is 1, and countable additivity.
2) Conditional probability is the probability of an event B given that event A has occurred, defined as P(B|A)=P(A∩B)/P(A). Events A and B are independent if P(B|A)=P(B) and P(A|B)=P(A).
3) Bayes' theorem gives the probability of an event A given that event B has occurred as P(A|B)=P(A)P(B|A)/P(B).
This document contains permissions and copyright information for Chapter 2 of the Handbook of Applied Cryptography. It grants permission to retrieve, print, and store a single copy of this chapter for personal use, but does not extend permission to bind multiple chapters, photocopy, produce additional copies, or make electronic copies available without prior written permission. Except as specifically permitted, the standard copyright from CRC Press applies and prohibits reproducing or transmitting the book or any part in any form without prior written permission.
This document discusses moment-generating functions and their properties and applications in probability theory. It defines the moment-generating function for both discrete and continuous random variables. Some key properties of moment-generating functions are that each probability distribution has a unique moment-generating function, and they can be used to find the distribution of sums of random variables. The document also describes several common probability distributions and derives their corresponding moment-generating functions.
Moment-Generating Functions and Reproductive Properties of DistributionsIJSRED
The document discusses moment-generating functions and their applications in probability theory. It defines the moment-generating function for both discrete and continuous random variables. Some key properties and examples of moment-generating functions for common distributions like binomial, Poisson, exponential, normal and gamma are provided. Reproductive properties of distributions are also described briefly.
The document discusses cumulative distribution functions (CDFs) and probability density functions (PDFs) for continuous random variables. It provides definitions and properties of CDFs and PDFs. For CDFs, it describes how they give the probability that a random variable is less than or equal to a value. For PDFs, it explains how they provide the probability of a random variable taking on a particular value. The document also gives examples of CDFs and PDFs for exponential and uniform random variables.
The document discusses different types of random variables including discrete, continuous, Bernoulli, binomial, geometric, Poisson, and uniform random variables. It provides the definitions and probability mass/density functions for each type. Examples are also given to illustrate concepts such as calculating probabilities for different random variables.
1. A random variable (RV) is a function that maps outcomes of a random phenomenon to numerical values. For a function X to be an RV, the set P(x) = {ω: X(ω) ≤ x} must be an event with a well-defined probability P for any x.
2. Conditional probability P(A|B) is the probability of event A occurring given that event B has occurred. It is defined as P(A ∩ B)/P(B).
3. The law of total probability states that for mutually exclusive and exhaustive events B1, B2, ..., the probability of event A is the sum of the probabilities of A given each Bi multiplied by
An Algorithm For The Combined Distribution And Assignment ProblemAndrew Parish
This document presents an algorithm for solving the combined distribution and assignment problem using generalized Benders' decomposition. The algorithm formulates the problem as a modified distribution problem with a minimax objective function instead of a linear one. It solves this master problem using the Newton-Kantorovich method for nonlinear concave programming problems with linear constraints. The algorithm iterates between solving the assignment problem given a distribution and solving the modified distribution problem subject to optimality constraints from the assignment problem. When the solution converges, it provides the optimal traffic flows for both distribution and assignment.
Chapter 1 - Probability Distributions.pdfMayurMandhub
This document outlines the key concepts covered in the Quantitative Finance 2 course. It discusses probability distributions, including defining random variables and their properties. It covers discrete and continuous probability distributions, and the use of probability density functions and cumulative distribution functions. The document also summarizes important probability rules like addition rule, multiplication rule, conditional probability, total probability rule, and Bayes' theorem. Sample problems and examples are provided to illustrate concepts like independent and dependent events, and how to calculate conditional and posterior probabilities.
The document discusses random processes and probability theory concepts relevant to communication systems. It defines key terms like random variables, sample space, events, probability, and distributions. It describes different types of random variables like discrete and continuous, and their probability mass functions and density functions. It also discusses statistical measures like mean, variance, and covariance that are used to characterize random signals and compare signals. Specific random variables discussed include binomial and uniform. The document provides foundations for analyzing random signals in communication systems.
This document provides an introduction to the author's work on developing a unified treatment of elliptic, hyperbolic, and partly elliptic-hyperbolic differential equations using symmetric positive linear operators. The author aims to show that these types of equations can be treated within a single framework by formulating the equations as systems of first-order equations and imposing admissible boundary conditions. Specifically:
1) The author introduces symmetric positive linear differential operators that satisfy certain algebraic properties and uses these to formulate differential equations as systems of first-order equations.
2) Admissible boundary conditions are defined that depend only on the nature of the operator coefficients on the boundary.
3) It is shown that under these conditions, the boundary value problems
Conditional mixture model and its application for regression modelLoc Nguyen
Expectation maximization (EM) algorithm is a powerful mathematical tool for estimating statistical parameter when data sample contains hidden part and observed part. EM is applied to learn finite mixture model in which the whole distribution of observed variable is average sum of partial distributions. Coverage ratio of every partial distribution is specified by the probability of hidden variable. An application of mixture model is soft clustering in which cluster is modeled by hidden variable whereas each data point can be assigned to more than one cluster and degree of such assignment is represented by the probability of hidden variable. However, such probability in traditional mixture model is simplified as a parameter, which can cause loss of valuable information. Therefore, in this research I propose a so-called conditional mixture model (CMM) in which the probability of hidden variable is modeled as a full probabilistic density function (PDF) that owns individual parameter. CMM aims to extend mixture model. I also propose an application of CMM which is called adaptive regression model (ARM). Traditional regression model is effective when data sample is scattered equally. If data points are grouped into clusters, regression model tries to learn a unified regression function which goes through all data points. Obviously, such unified function is not effective to evaluate response variable based on grouped data points. The concept “adaptive” of ARM means that ARM solves the ineffectiveness problem by selecting the best cluster of data points firstly and then evaluating response variable within such best cluster. In orther words, ARM reduces estimation space of regression model so as to gain high accuracy in calculation.
Keywords: expectation maximization (EM) algorithm, finite mixture model, conditional mixture model, regression model, adaptive regression model (ARM).
Noise is unwanted sound considered unpleasant, loud, or disruptive to hearing. From a physics standpoint, there is no distinction between noise and desired sound, as both are vibrations through a medium, such as air or water. The difference arises when the brain receives and perceives a sound.
Error Estimates for Multi-Penalty Regularization under General Source Conditioncsandit
In learning theory, the convergence issues of the regression problem are investigated with
the least square Tikhonov regularization schemes in both the RKHS-norm and the L 2
-norm.
We consider the multi-penalized least square regularization scheme under the general source
condition with the polynomial decay of the eigenvalues of the integral operator. One of the
motivation for this work is to discuss the convergence issues for widely considered manifold
regularization scheme. The optimal convergence rates of multi-penalty regularizer is achieved
in the interpolation norm using the concept of effective dimension. Further we also propose
the penalty balancing principle based on augmented Tikhonov regularization for the choice of
regularization parameters. The superiority of multi-penalty regularization over single-penalty
regularization is shown using the academic example and moon data set.
a) Use Newton’s Polynomials for Evenly Spaced data to derive the O(h.pdfpetercoiffeur18
a) Use Newton’s Polynomials for Evenly Spaced data to derive the O(h4) accurate Second
Centered Difference approximation of the 1st derivative at nx. Start with a polynomial fit to
points at n-2x , n-1x, nx , n+1x and n+2x .
b) Use Newton’s Polynomials for Evenly Spaced data to derive the O(h4) accurate Second
Centered Difference approximation of the 2nd derivative at nx . Remember, to keep the same
O(h4) accuracy, while taking one more derivative than in Part a, we need to add a point to the
polynomial we used in part a.t,s01530456075y,km0356488107120
Solution
An interpolation assignment generally entails a given set of information points: in which the
values yi can,
xi x0 x1 ... xn
f(xi) y0 y1 ... yn
for instance, be the result of a few bodily measurement or they can come from a long
numerical calculation. hence we know the fee of the underlying characteristic f(x) at the set
of points xi, and we want to discover an analytic expression for f .
In interpolation, the assignment is to estimate f(x) for arbitrary x that lies among the smallest
and the most important xi
. If x is out of doors the variety of the xi’s, then the task is called extrapolation,
which is substantially greater unsafe.
with the aid of far the maximum not unusual useful paperwork utilized in interpolation are the
polynomials.
different picks encompass, as an instance, trigonometric functions and spline features
(mentioned
later during this direction).
Examples of different sorts of interpolation responsibilities include:
1. Having the set of n + 1 information factors xi
, yi, we want to understand the fee of y in the
complete c program languageperiod x = [x0, xn]; i.e. we need to find a simple formulation
which reproduces
the given points exactly.
2. If the set of statistics factors contain errors (e.g. if they are measured values), then we
ask for a components that represents the records, and if feasible, filters out the errors.
3. A feature f may be given within the shape of a pc system which is high priced
to assess. In this case, we want to find a characteristic g which offers a very good
approximation of f and is simpler to assess.
2 Polynomial interpolation
2.1 Interpolating polynomial
Given a fixed of n + 1 records points xi
, yi, we need to discover a polynomial curve that passes
via all the factors. as a consequence, we search for a non-stop curve which takes at the values yi
for every of the n+1 wonderful xi’s.
A polynomial p for which p(xi) = yi whilst zero i n is stated to interpolate the given set of
records points. The factors xi are known as nodes.
The trivial case is n = zero. right here a steady function p(x) = y0 solves the hassle.
The only case is n = 1. In this situation, the polynomial p is a directly line described via
p(x) =
xx1
x0 x1
y0 +
xx0
x1 x0
y1
= y0 +
y1 y0
x1 x0
(xx0)
here p is used for linear interpolation.
As we will see, the interpolating polynomial may be written in an expansion of paperwork,
among
these are the Newton shape and the Lag.
This document discusses using the Newton-Raphson iterative method to solve chemical equilibrium problems. It begins by introducing fixed point theory and the Newton-Raphson method for solving nonlinear equations. It then describes applying this method to determine the O reactant ratio that produces an adiabatic equilibrium temperature in the chemical reaction of partial methane oxidation. Specifically, it develops a system of seven nonlinear equations and uses the Newton-Raphson method to iteratively solve for the fixed point and desired chemical equilibrium conditions.
On Analytic Review of Hahn–Banach Extension Results with Some GeneralizationsBRNSS Publication Hub
The useful Hahn–Banach theorem in functional analysis has significantly been in use for many years ago. At this point in time, we discover that its domain and range of existence can be extended point wisely so as to secure a wider range of extendibility. In achieving this, we initially reviewed the existing traditional Hahn–Banach extension theorem, before we carefully and successfully used it to generate the finite extension form as in main results of section three.
This document provides an overview and proofs of several theorems related to the Hahn-Banach theorem. It begins with an introduction to linear functionals and the Hahn-Banach theorem. It then presents two main theorems - the Hahn-Banach theorem and the topological Hahn-Banach theorem. The document provides proofs of these theorems and several related theorems using the Hahn-Banach extension lemma. It also discusses consequences of the Hahn-Banach extension form and provides proofs of the theorems using the lemma.
This document provides an overview and definitions related to functional analysis and Banach spaces. It discusses:
1) The definition of a Banach space as a complete normed linear space where every Cauchy sequence converges.
2) Examples of Banach spaces including l^p spaces, C(X) for compact X, C_b(X) for any topological space X, and C^k([a,b]).
3) Measure spaces and the definition of measurable functions on a measure space. It notes the closure properties of measurable functions under scalar multiplication and (sometimes) addition.
Leveraging Generative AI to Drive Nonprofit InnovationTechSoup
In this webinar, participants learned how to utilize Generative AI to streamline operations and elevate member engagement. Amazon Web Service experts provided a customer specific use cases and dived into low/no-code tools that are quick and easy to deploy through Amazon Web Service (AWS.)
This presentation was provided by Rebecca Benner, Ph.D., of the American Society of Anesthesiologists, for the second session of NISO's 2024 Training Series "DEIA in the Scholarly Landscape." Session Two: 'Expanding Pathways to Publishing Careers,' was held June 13, 2024.
Chapter wise All Notes of First year Basic Civil Engineering.pptxDenish Jangid
Chapter wise All Notes of First year Basic Civil Engineering
Syllabus
Chapter-1
Introduction to objective, scope and outcome the subject
Chapter 2
Introduction: Scope and Specialization of Civil Engineering, Role of civil Engineer in Society, Impact of infrastructural development on economy of country.
Chapter 3
Surveying: Object Principles & Types of Surveying; Site Plans, Plans & Maps; Scales & Unit of different Measurements.
Linear Measurements: Instruments used. Linear Measurement by Tape, Ranging out Survey Lines and overcoming Obstructions; Measurements on sloping ground; Tape corrections, conventional symbols. Angular Measurements: Instruments used; Introduction to Compass Surveying, Bearings and Longitude & Latitude of a Line, Introduction to total station.
Levelling: Instrument used Object of levelling, Methods of levelling in brief, and Contour maps.
Chapter 4
Buildings: Selection of site for Buildings, Layout of Building Plan, Types of buildings, Plinth area, carpet area, floor space index, Introduction to building byelaws, concept of sun light & ventilation. Components of Buildings & their functions, Basic concept of R.C.C., Introduction to types of foundation
Chapter 5
Transportation: Introduction to Transportation Engineering; Traffic and Road Safety: Types and Characteristics of Various Modes of Transportation; Various Road Traffic Signs, Causes of Accidents and Road Safety Measures.
Chapter 6
Environmental Engineering: Environmental Pollution, Environmental Acts and Regulations, Functional Concepts of Ecology, Basics of Species, Biodiversity, Ecosystem, Hydrological Cycle; Chemical Cycles: Carbon, Nitrogen & Phosphorus; Energy Flow in Ecosystems.
Water Pollution: Water Quality standards, Introduction to Treatment & Disposal of Waste Water. Reuse and Saving of Water, Rain Water Harvesting. Solid Waste Management: Classification of Solid Waste, Collection, Transportation and Disposal of Solid. Recycling of Solid Waste: Energy Recovery, Sanitary Landfill, On-Site Sanitation. Air & Noise Pollution: Primary and Secondary air pollutants, Harmful effects of Air Pollution, Control of Air Pollution. . Noise Pollution Harmful Effects of noise pollution, control of noise pollution, Global warming & Climate Change, Ozone depletion, Greenhouse effect
Text Books:
1. Palancharmy, Basic Civil Engineering, McGraw Hill publishers.
2. Satheesh Gopi, Basic Civil Engineering, Pearson Publishers.
3. Ketki Rangwala Dalal, Essentials of Civil Engineering, Charotar Publishing House.
4. BCP, Surveying volume 1
Level 3 NCEA - NZ: A Nation In the Making 1872 - 1900 SML.pptHenry Hollis
The History of NZ 1870-1900.
Making of a Nation.
From the NZ Wars to Liberals,
Richard Seddon, George Grey,
Social Laboratory, New Zealand,
Confiscations, Kotahitanga, Kingitanga, Parliament, Suffrage, Repudiation, Economic Change, Agriculture, Gold Mining, Timber, Flax, Sheep, Dairying,
How Barcodes Can Be Leveraged Within Odoo 17Celine George
In this presentation, we will explore how barcodes can be leveraged within Odoo 17 to streamline our manufacturing processes. We will cover the configuration steps, how to utilize barcodes in different manufacturing scenarios, and the overall benefits of implementing this technology.
Philippine Edukasyong Pantahanan at Pangkabuhayan (EPP) CurriculumMJDuyan
(𝐓𝐋𝐄 𝟏𝟎𝟎) (𝐋𝐞𝐬𝐬𝐨𝐧 𝟏)-𝐏𝐫𝐞𝐥𝐢𝐦𝐬
𝐃𝐢𝐬𝐜𝐮𝐬𝐬 𝐭𝐡𝐞 𝐄𝐏𝐏 𝐂𝐮𝐫𝐫𝐢𝐜𝐮𝐥𝐮𝐦 𝐢𝐧 𝐭𝐡𝐞 𝐏𝐡𝐢𝐥𝐢𝐩𝐩𝐢𝐧𝐞𝐬:
- Understand the goals and objectives of the Edukasyong Pantahanan at Pangkabuhayan (EPP) curriculum, recognizing its importance in fostering practical life skills and values among students. Students will also be able to identify the key components and subjects covered, such as agriculture, home economics, industrial arts, and information and communication technology.
𝐄𝐱𝐩𝐥𝐚𝐢𝐧 𝐭𝐡𝐞 𝐍𝐚𝐭𝐮𝐫𝐞 𝐚𝐧𝐝 𝐒𝐜𝐨𝐩𝐞 𝐨𝐟 𝐚𝐧 𝐄𝐧𝐭𝐫𝐞𝐩𝐫𝐞𝐧𝐞𝐮𝐫:
-Define entrepreneurship, distinguishing it from general business activities by emphasizing its focus on innovation, risk-taking, and value creation. Students will describe the characteristics and traits of successful entrepreneurs, including their roles and responsibilities, and discuss the broader economic and social impacts of entrepreneurial activities on both local and global scales.
THE SACRIFICE HOW PRO-PALESTINE PROTESTS STUDENTS ARE SACRIFICING TO CHANGE T...indexPub
The recent surge in pro-Palestine student activism has prompted significant responses from universities, ranging from negotiations and divestment commitments to increased transparency about investments in companies supporting the war on Gaza. This activism has led to the cessation of student encampments but also highlighted the substantial sacrifices made by students, including academic disruptions and personal risks. The primary drivers of these protests are poor university administration, lack of transparency, and inadequate communication between officials and students. This study examines the profound emotional, psychological, and professional impacts on students engaged in pro-Palestine protests, focusing on Generation Z's (Gen-Z) activism dynamics. This paper explores the significant sacrifices made by these students and even the professors supporting the pro-Palestine movement, with a focus on recent global movements. Through an in-depth analysis of printed and electronic media, the study examines the impacts of these sacrifices on the academic and personal lives of those involved. The paper highlights examples from various universities, demonstrating student activism's long-term and short-term effects, including disciplinary actions, social backlash, and career implications. The researchers also explore the broader implications of student sacrifices. The findings reveal that these sacrifices are driven by a profound commitment to justice and human rights, and are influenced by the increasing availability of information, peer interactions, and personal convictions. The study also discusses the broader implications of this activism, comparing it to historical precedents and assessing its potential to influence policy and public opinion. The emotional and psychological toll on student activists is significant, but their sense of purpose and community support mitigates some of these challenges. However, the researchers call for acknowledging the broader Impact of these sacrifices on the future global movement of FreePalestine.
A Visual Guide to 1 Samuel | A Tale of Two HeartsSteve Thomason
These slides walk through the story of 1 Samuel. Samuel is the last judge of Israel. The people reject God and want a king. Saul is anointed as the first king, but he is not a good king. David, the shepherd boy is anointed and Saul is envious of him. David shows honor while Saul continues to self destruct.
1. Conditional Probability Mass Function
• The probability distribution of a discrete random variable can be characterized by its probability mass function.
• When the probability distribution of the random variable is updated, in order to consider some information that
gives rise to a conditional probability distribution, then such a conditional distribution can be characterized by a
conditional probability mass function.
• Conditional PMF is especially appropriate when the experiment is a compound one, in which the second part of
the experiment depends upon the outcome of the first part.
• It has the usual properties of a PMF, that of being between 0 and 1 and also summing to one.
2. Conditional Probability Mass Function
• We recall that a conditional probability P[AIB] is the probability of an event A, given that we know that some
other event B has occurred.
• Except for the case when the two events are independent of each other , the knowledge that B has occurred will
change the probability P[A]. In other words, P[AIB] is our new probability in light of the additional knowledge.
• In many practical situations, two random mechanisms are at work and are described by events A and B .
• To compute probabilities for a complex experiment it is usually convenient to use a conditioning argument to
simplify the reasoning.
• For example, say we choose one of two coins and toss it 4 times. We might inquire as to the probability of
observing 2 or more heads. However, this probability will depend upon which coin was chosen, as for example in
the situation where one coin is fair and the other coin is weighted.
• It is therefore convenient to define conditional probability mass functions, Px [klcoin 1 chosen] and Px[klcoin 2
chosen], since once we know which coin is chosen, we can easily specify the PMF.
• In particular, for this example the conditional PMF is a binomial one whose value of P depends upon which coin is
chosen and with k denoting the number of heads. Once the conditional PMFs are known, we have by the law of
total probability that the probability of observing k heads for this experiment is given by the PMF:
Px [k] = Px [klcoin 1 chosen]P[coin 1 chosen] + Px[klcoin 2 chosen]P[coin 2 chosen]
3. Conditional Probability Mass Function cont.
• The PMF that is required depends directly on the conditional PMFs (of which there are two).
• The use of conditional PMFs greatly simplifies our task in that given the event, i.e., the coin chosen, the PMF of
the number of heads observed readily follows. Also, in many problems, including this one, it is actually the
conditional PMFs that are specified in the description of the experimental procedure.
• It makes sense, therefore, to define a conditional PMF and study its properties.
• For the most part, the definitions and properties will mirror those of the conditional probability P[AIB], where A
and B are events defined on SX,Y.
• Just as we used conditional probabilities to evaluate the likelihood of one event given another, we develop here
the concepts of continuous conditional distributions and continuous conditional probability mass functions and
probability density functions to evaluate the behavior of one random variable given knowledge of another.
• Conditional probability P[A|B] is a number that expresses our new knowledge about the occurrence of event A,
when we learn that another event B occurs.
• In Conditional Probability Mass Function we consider event A to be the observation of a particular value of a
random variable: A = {X = x}.
• The conditioning event B contains information about X but not the precise value of X.
• If X ≤ 33 we learn of the occurrence of an event B that describes some property of X. The occurrence of the
conditioning event B changes the probabilities of the event {X = x}.
4. Conditional Probability Mass Function cont.
We can find the conditional probabilities P [A|B] = P [X = x|B] for all real numbers x. This collection of probabilities is
a function of x.
It is the conditional probability mass function of random variable X, given that B occurred.
Definition 2.19 Conditional PMF
Given the event B, with P[B] > 0, the conditional probability mass function of X is
PX|B (x) = P [X = x|B]
About notation:
The name of a PMF is the letter P with a subscript containing the name of the random variable. For a conditional
PMF, the subscript contains the name of the random variable followed by a vertical bar followed by a statement of
the conditioning event.
The argument of the function is usually the lowercase letter corresponding to the variable name. The argument is a
dummy variable. It could be any letter, so that PX|B(x) is the same function as PX|B(u).
Sometimes we write the function with no specified argument at all, PX|B(·).
In some applications, we begin with a set of conditional PMFs, PX|Bi(x), i = 1, 2, . . ,m, where B1, B2, . . . , Bm is an event
space.
We then use the law of total probability to find the PMF PX (x):
P [A] = ∑i=1,m P [A|Bi ] P [Bi ]
where: P(A|B) = P(A ∩ B) / P(B)
5. Conditional Probability Mass Function cont.
Theorem 2.16
A random variable X resulting from an experiment with event space B1, B2, . . . , Bm has
PMF > PX (x) = ∑i=1,m PX|Bi (x) P [Bi ]
Proof: The theorem follows directly from Theorem 1.10 (Law of Total Probability)
( P [A] = ∑i=1,m P [A|Bi ] P [Bi ] ) with A denoting the event {X = x} )
When a conditioning event B ⊂ SX , the PMF PX (x) determines both the probability of B as well as the conditional
PMF:
PX|B (x) = P [X = x, B] / P [B]
Now either the event X = x is contained in the event B or it is not.
If x ∈ B, then {X = x}∩ B = {X = x} and P[X = x, B] = PX (x). Otherwise, if x ∈ B, then {X = x}∩ B = ∅ and P[X = x, B] = 0.
The next theorem uses Equation PX|B (x) = P [X = x, B] / P [B] to calculate the conditional PMF.
6. Conditional Probability Mass Function cont.
Theorem 2.17
PX|B (x) =
PX(x)
P[B]
, x∈B
0, otherwise
The theorem states that when we learn that an outcome x∈B, the probabilities of all x ∉ B are zero in our conditional
model and the probabilities of all x∈B are proportionally higher than they were before we learned x ∈ B.
PX|B(x) is a perfectly respectable PMF.
Because the conditioning event B tells us that all possible outcomes are in B, we rewrite Theorem 2.1. which is:
For a discrete random variable X with PMF PX (x) and range SX :
(a) For any x, PX(x) ≥ 0
(b) ∑x∈SX PX (x) = 1
(c) For any event B ⊂ SX , the probability that X is in the set B is P[B] =∑x∈B PX(x)
using B in place of S, we have next Theorem.
7. Conditional Probability Mass Function cont.
Theorem 2.18
(a) For any x ∈ B, PX|B(x) ≥ 0
(b) ∑x∈B PX|B (x) = 1
(c) For any event C ⊂ B, P[C|B], the conditional probability that X is in the set C, is P[C|B] =∑x∈C PX|B (x)
Therefore, we can compute averages of the conditional random variable X|B and averages of functions of X|B in the
same way that we compute averages of X.
The only difference is that we use the conditional PMF PX|B(·) in place of PX (·)
8. Conditional Probability Mass Function cont.
Definition 2.20
Conditional Expected Value
The conditional expected value of random variable X given condition B is
E [X|B] = μX|B =∑x∈B x PX|B(x)
When we are given a family of conditional probability models PX|Bi (x) for an event space B1, B2, . . . , Bm, we can
compute the expected value E[X] in terms of the conditional expected values E[X|Bi]
Theorem 2.19 For a random variable X resulting from an experiment with event space B1, B2, . . . , Bm,
E [X] = ∑i=1,m E [X|Bi ] P [Bi ]
Proof
Since E[X] = ∑x x PX (x), we can use (Theorem 2.16 ) PX (x) = ∑i=1,m PX|Bi (x) P [Bi ]
to write
E [X] = ∑x x∑i=1,m PX|Bi (x) P [Bi ] =∑i=1,m P [Bi ] ∑x xPX|Bi (x) = ∑i=1,m P [Bi ] E [X|Bi ]
9. Conditional Probability Mass Function cont.
For a derived random variable Y = g(X), we have the equivalent of (Theorem 2.10):
E [Y ] = μY = ∑x∈SX
g(x) PX (x)
Theorem 2.20 The conditional expected value of Y = g(X) given condition B is
E [Y | B] = E [g(X) | B] = ∑x∈B g(x) PX|B (x)
The function g(Xi) = EYlx[Ylxi] is the mean of the conditional PMF PYlx[Yjlxi]. Alternatively, it is known as the conditional
mean.
This terminology is widespread and so we will adhere to it , although we should keep in mind that it is meant to
denote the usual mean of the conditional PMF.
It is also of interest to determine the expectation of other quantities besides Y with respect to the conditional PMF.
This is called the conditional expectation and is symbolized by EYlx[g(Y)lxi].
The latter is called the conditional expectation of g(Y). For example, if g(Y) = y2, then it becomes the conditional
expectation of y2 or equivalently the conditional second moment.
Lastly, the we should be aware that the conditional mean is the optimal predictor of a random variable based on
observation of a second random variable.
10. Conditional Probability Mass Function cont.
It follows that the conditional variance and conditional standard deviation conform to Definitions 2.16
Var (X) = E [(X- μX)2]
and Definitions 2.17
𝜎𝑋= 𝑉𝑎𝑟[𝑋] with X|B replacing X:
𝜎X|B
= 𝑉𝑎𝑟[X|B]
To conclude: The utility of defining a conditional PMF is especially appropriate when the experiment is a compound
one, in which the second part of the experiment depends upon the outcome of the first part.
The definition of the conditional PMF has the usual properties of a PMF, that of being between 0 and 1 and also
summing to one.
11. Chapter Summary
With all of the concepts and formulas introduced in this chapter, there is a high probability that the beginning
student will be confused at this point.
Part of the problem is that we are dealing with several different mathematical entities including random variables,
probability functions, and parameters.
Before plugging numbers or symbols into a formula, it is good to know what the entities are.
The random variable X transforms outcomes of an experiment to real numbers. Note that X is the name of the
random variable.
A possible observation is x, which is a number. SX is the range of X, the set of all possible observations x.
The PMF PX (x) is a function that contains the probability model of the random variable X.
The PMF gives the probability of observing any x. PX (·) contains our information about the randomness of X.
12. Chapter Summary
The expected value E[X] = μX and the variance Var[X] are numbers that describe the entire probability model.
Mathematically, each is a property of the PMF PX (·). The expected value is a typical value of the random variable.
The variance describes the dispersion of sample values about the expected value.
A function of a random variable Y = g(X) transforms the random variable X into a different random variable Y .
For each observation X = x, g(·) is a rule that tells you how to calculate y = g(x), a sample value of Y .
Although PX (·) and g(·) are both mathematical functions, they serve different purposes here. PX (·) describes the
randomness in an experiment.
On the other hand, g(·) is a rule for obtaining a new random variable from a random variable you have observed.
The Conditional PMF PX|B(x) is the probability model that we obtain when we gain partial knowledge of the outcome
of an experiment.
The partial knowledge is that the outcome x ∈ B ⊂ SX . The conditional probability model has its own expected value,
E[X|B], and its own variance, Var[X|B].