TrueMotion Data Science Lunch Seminar for September 19, 2016, wherein we discuss the theory behind A/B testing and some best practices for its real-world application.
Pengolahan Data Panel Logit di Stata: Penilaian Goodness of Fit, Uji Model, d...The1 Uploader
panel data regression in stata with tests of classical assumptions violation
goodness of fit
pooled least square
fixed effect
random effect
logit
panel logit
1. The document discusses various methods for continuous optimization, including rates of convergence for noise-free and noisy settings.
2. In noise-free settings, methods like Newton's method and BFGS have quadratic or superlinear convergence rates, while evolutionary strategies (ES) have linear convergence rates.
3. Lower bounds on optimization complexity are also discussed, showing minimum comparisons or evaluations needed depending on problem properties like domain size and precision required.
This document discusses Boolean algebra properties and concepts. It defines Boolean variables and constants, and lists common Boolean algebra properties like idempotence, complement, duality, commutativity, associativity, distributivity, De Morgan's laws, and absorption. It also discusses Boolean expressions, truth tables, canonical forms, minterms, maxterms, and how to derive the complement and canonical forms of a Boolean function.
Using R Tool for Probability and Statistics nazlitemu
1. The document describes exercises from a probability and statistics lab report, including generating random vectors, estimating distributions, and assessing hypotheses.
2. For the first exercise, random vectors were generated from uniform, normal, and exponential distributions and their histograms, CDFs, and boxplots were represented. Bin sizes were also calculated.
3. Subsequent exercises involved comparing mean and variance, assessing dependence between random variables, modeling loss event data, and applying the central limit theorem.
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.
The document describes functions in Python. It defines what a function is, how to define and invoke functions, and how functions work. It discusses function headers, parameters, arguments, return values, and function bodies. Examples are provided to demonstrate defining a max() function to return the maximum of two numbers, and testing/invoking the function. The use of functions to organize code and enable reuse is discussed.
The document describes exercises on structures and strings in C programming. It defines a structure called _PLAYER to store details of players like name, date of birth, height and weight. It then shows how to define the structure, read and print records of multiple players from an array of _PLAYER structures. Functions are defined to find the tallest player, print names in descending order of height, check character cases, convert case of characters in a string, and compute operations on structures like distance between points.
The Ring programming language version 1.4 book - Part 21 of 30Mahmoud Samir Fayed
This document provides information about low level functions in Ring that interface with C and the Ring virtual machine. It discusses the callgc() function to manually call the garbage collector, the varptr() function to get a pointer to a C variable, the space() function to allocate memory, nullpointer() to pass a null pointer, object2pointer() and pointer2object() to convert between Ring objects and C pointers, ptrcmp() to compare pointers, and functions like ringvm_cfunctionslist() to get lists of C functions and ringvm_functionslist() to get a list of Ring functions from the bytecode. These low level functions provide interfaces between Ring and C to integrate Ring with C libraries and code.
Pengolahan Data Panel Logit di Stata: Penilaian Goodness of Fit, Uji Model, d...The1 Uploader
panel data regression in stata with tests of classical assumptions violation
goodness of fit
pooled least square
fixed effect
random effect
logit
panel logit
1. The document discusses various methods for continuous optimization, including rates of convergence for noise-free and noisy settings.
2. In noise-free settings, methods like Newton's method and BFGS have quadratic or superlinear convergence rates, while evolutionary strategies (ES) have linear convergence rates.
3. Lower bounds on optimization complexity are also discussed, showing minimum comparisons or evaluations needed depending on problem properties like domain size and precision required.
This document discusses Boolean algebra properties and concepts. It defines Boolean variables and constants, and lists common Boolean algebra properties like idempotence, complement, duality, commutativity, associativity, distributivity, De Morgan's laws, and absorption. It also discusses Boolean expressions, truth tables, canonical forms, minterms, maxterms, and how to derive the complement and canonical forms of a Boolean function.
Using R Tool for Probability and Statistics nazlitemu
1. The document describes exercises from a probability and statistics lab report, including generating random vectors, estimating distributions, and assessing hypotheses.
2. For the first exercise, random vectors were generated from uniform, normal, and exponential distributions and their histograms, CDFs, and boxplots were represented. Bin sizes were also calculated.
3. Subsequent exercises involved comparing mean and variance, assessing dependence between random variables, modeling loss event data, and applying the central limit theorem.
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.
The document describes functions in Python. It defines what a function is, how to define and invoke functions, and how functions work. It discusses function headers, parameters, arguments, return values, and function bodies. Examples are provided to demonstrate defining a max() function to return the maximum of two numbers, and testing/invoking the function. The use of functions to organize code and enable reuse is discussed.
The document describes exercises on structures and strings in C programming. It defines a structure called _PLAYER to store details of players like name, date of birth, height and weight. It then shows how to define the structure, read and print records of multiple players from an array of _PLAYER structures. Functions are defined to find the tallest player, print names in descending order of height, check character cases, convert case of characters in a string, and compute operations on structures like distance between points.
The Ring programming language version 1.4 book - Part 21 of 30Mahmoud Samir Fayed
This document provides information about low level functions in Ring that interface with C and the Ring virtual machine. It discusses the callgc() function to manually call the garbage collector, the varptr() function to get a pointer to a C variable, the space() function to allocate memory, nullpointer() to pass a null pointer, object2pointer() and pointer2object() to convert between Ring objects and C pointers, ptrcmp() to compare pointers, and functions like ringvm_cfunctionslist() to get lists of C functions and ringvm_functionslist() to get a list of Ring functions from the bytecode. These low level functions provide interfaces between Ring and C to integrate Ring with C libraries and code.
The document discusses linear regression and maximum likelihood estimation in EViews. It provides an overview of the linear regression model and ordinary least squares (OLS) estimation. It then discusses how to perform OLS regression and test linear restrictions in EViews. Finally, it introduces maximum likelihood estimation and how it provides an alternative framework for estimation when the data is assumed to follow a particular probability distribution.
Building Machine Learning Algorithms on Apache Spark with William BentonSpark Summit
There are lots of reasons why you might want to implement your own machine learning algorithms on Spark: you might want to experiment with a new idea, try and reproduce results from a recent research paper, or simply to use an existing technique that isn’t implemented in MLlib. In this talk, we’ll walk through the process of developing a new machine learning model for Spark. We’ll start with the basics, by considering how we’d design a parallel implementation of a particular unsupervised learning technique. The bulk of the talk will focus on the details you need to know to turn an algorithm design into an efficient parallel implementation on Spark: we’ll start by reviewing a simple RDD-based implementation, show some improvements, point out some pitfalls to avoid, and iteratively extend our implementation to support contemporary Spark features like ML Pipelines and structured query processing. You’ll leave this talk with everything you need to build a new machine learning technique that runs on Spark.
This document contains C++ code examples using numerical methods like Newton-Raphson, successive approximation, and secant methods to find the real roots of various equations. It includes 6 questions/problems with sample code to find roots of equations like x^4 - 11x = 8, e^x - 3x^2 = 0, x^3 - 2x - 3 = 0, and sin(x) + 3cos(x) = 2. For each problem, the code provided the numerical method used, input/output examples, and calculations to iteratively approximate the root to within a set tolerance.
The Ring programming language version 1.3 book - Part 59 of 88Mahmoud Samir Fayed
The document provides details about printing the final intermediate code generated after executing a Ring program.
It begins by showing the output of running a test Ring program and printing the bytecode. This includes the byte code instructions, operation codes, program counter, data, and other details.
It then shows sections of the large intermediate code output, including function and method definitions, variable assignments, calls between functions, and other low-level operations.
The document explains that the output provides a detailed view of the final bytecode generated from the Ring source code after it has been executed by the Ring virtual machine. This allows viewing and understanding the low-level operations performed during program execution.
Super TypeScript II Turbo - FP Remix (NG Conf 2017)Sean May
This talk focuses on typical functional programming paradigms in JavaScript, as implemented in TypeScript.
The goal of this talk was to provide common ground in FP paradigms, between C# .NET developers, Java Spring developers and JS programmers. The slides have been annotated and extended from the talk, to cover intended concepts not explicit in the code examples, themselves.
https://www.youtube.com/watch?v=9oVKjZrgXmU
The web is evolving, we got it. One of the clear consequences is the complexity of our web apps (formerly known as ‘websites’). The conciseness of functional programming and its fundamentals got our attention, but we knew we could do better. And now we have the Reactive programming model, a functional and declarative way of dealing with big amounts of data.
In the center of it we have Observables: objects responsible to keep your application alive, reacting to any mutation your data may have, through any period of time. We’ll take a look on the concepts and also on the lib that implements it in Angular’s core: RxJS. Using the provided operators, we have great power on our hands, doing anything imaginable in a concise, declarative and easy-to-maintain way.
Watch out: observables are here to stay!
The Ring programming language version 1.5.2 book - Part 74 of 181Mahmoud Samir Fayed
This document provides information about low level functions in Ring that allow interaction with C code and pointers. It describes the following functions:
- callgc() forces garbage collection to free temporary variables
- varptr() gets a C pointer for a Ring variable
- space() allocates memory and returns a string pointer
- nullpointer() returns a NULL pointer for optional parameters
- object2pointer() gets a C pointer for a Ring list or object
- pointer2object() converts a C pointer back to a Ring list or object
- ptrcmp() compares two C pointers
It also briefly mentions ringvm functions for getting information about the runtime environment like loaded classes, functions, memory usage and more.
This document defines key probability terms and concepts. It begins by defining a probability experiment, outcomes, sample space, events such as simple, compound and null events. It then discusses union, intersection and complements of events. It also defines equally likely outcomes, mutually exclusive events, exhaustive events, and conditional probability and independence. The document provides examples to illustrate these concepts and definitions. It concludes by discussing approaches to measure probability such as the classical, relative frequency, subjective and axiomatic approaches. It also covers rules of probability including addition, multiplication and conditional probability rules.
We are a growth marketing agency, that helps startups and well establish companies to achieve rapid and sustainable growth. We use various types of marketing and product iterations — content marketing, social media marketing, paid ads, email marketing, SEO and viral strategies, among others, with a purpose to increase the conversion rate and achieve rapid growth of the user base.
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.
1) The document discusses the principle of mathematical induction, which is used to prove that a proposition P(n) is true for all natural numbers n. It involves showing that P(0) is true, and if P(n) is true then P(n+1) is true, which implies P(n) is true for all n.
2) An example uses induction to prove that n < 2n for all positive integers n. It shows the basis step P(1) is true, and inductive step that if P(n) is true then P(n+1) is true.
3) Another example uses induction to prove the summation formula 1 + 2 + ... +
1) The document discusses the principle of mathematical induction, which is used to prove that a proposition P(n) is true for all natural numbers n. It involves showing that P(0) is true, and if P(n) is true then P(n+1) is true, which implies P(n) is true for all n.
2) An example uses induction to prove that n < 2n for all positive integers n. It shows the basis step P(1) is true, and inductive step that if P(n) is true then P(n+1) is true.
3) Another example uses induction to prove the summation formula 1 + 2 + ... +
The simplest and most common form of mathematical induction infers that a statement involving a natural number n (that is, an integer n ≥ 0 or 1) holds for all values of n. The proof consists of two steps:
The base case (or initial case): prove that the statement holds for 0, or 1.
The induction step (or inductive step, or step case): prove that for every n, if the statement holds for n, then it holds for n + 1. In other words, assume that the statement holds for some arbitrary natural number n, and prove that the statement holds for n + 1
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.
1) The document provides four probability problems involving combinations and permutations to further understanding of combinatorics. It gives the solutions and explanations for each problem involving probabilities of card hands.
2) It then introduces conditional probability and uses examples like the Monty Hall problem to illustrate how conditioning on additional information can change a probability. It provides the definition of conditional probability and proves Bayes' theorem.
3) The document discusses independence of events and uses examples like coin flips and natural disasters to demonstrate independence. It also introduces the continuity of probability and uses examples like the Cantor set to illustrate how it allows calculating probabilities of infinite sets.
The simplest and most common form of mathematical induction infers that a statement involving a natural number n (that is, an integer n ≥ 0 or 1) holds for all values of n. The proof consists of two steps:
The base case (or initial case): prove that the statement holds for 0, or 1.
The induction step (or inductive step, or step case): prove that for every n, if the statement holds for n, then it holds for n + 1. In other words, assume that the statement holds for some arbitrary natural number n, and prove that the statement holds for n + 1
This document contains lecture notes on reliability engineering. It covers basic probability theory concepts like probability distributions, random variables, and rules for combining probabilities. It then discusses reliability topics like definitions of reliability, hazard rate, and measures of reliability like mean time to failure. It also covers classifications of engineering systems into series, parallel and other configurations and how to evaluate their reliability. Finally, it discusses discrete and continuous Markov chains and how to model repairable systems using these techniques.
This document discusses mathematical induction, which is a method for proving that a proposition is true for all natural numbers. It involves showing that the proposition is true for the base case, usually 0 or 1, and assuming the proposition is true for some value n to prove it is also true for n+1. Two examples are provided: proving n < 2n for all positive integers n, and proving the formula for summing the first n positive integers. The document also introduces the second principle of mathematical induction, which involves showing the proposition is true for values up to n before proving it for n+1.
This document provides definitions and formulas for key concepts in descriptive statistics, probability, and common probability distributions including:
- Descriptive statistics such as mean, median, mode, variance, and standard deviation.
- Probability concepts such as probability, events, unions/intersections of events, and basic counting rules.
- Common probability distributions like the binomial, uniform, and normal distributions along with their expected values, variances, and probabilities. Formulas for transformations are also included.
The document is intended as a reference sheet for statistics concepts and calculations in a concise format.
Talk at CIRM on Poisson equation and debiasing techniquesPierre Jacob
- The document discusses debiasing techniques for Markov chain Monte Carlo (MCMC) algorithms.
- It introduces the concept of "fishy functions" which are solutions to Poisson's equation and can be used for control variates to reduce bias and variance in MCMC estimators.
- The document outlines different sections including revisiting unbiased estimation through Poisson's equation, asymptotic variance estimation using a novel "fishy function" estimator, and experiments on different examples.
The document summarizes key concepts about the binomial and geometric distributions:
The binomial distribution models the number of successes in a fixed number of yes/no trials where the probability of success is constant. The geometric distribution models the number of trials until the first success. Both have calculators functions and follow patterns for mean, standard deviation, and normal approximations. Formulas for probability mass and cumulative distribution functions are provided.
The document discusses linear regression and maximum likelihood estimation in EViews. It provides an overview of the linear regression model and ordinary least squares (OLS) estimation. It then discusses how to perform OLS regression and test linear restrictions in EViews. Finally, it introduces maximum likelihood estimation and how it provides an alternative framework for estimation when the data is assumed to follow a particular probability distribution.
Building Machine Learning Algorithms on Apache Spark with William BentonSpark Summit
There are lots of reasons why you might want to implement your own machine learning algorithms on Spark: you might want to experiment with a new idea, try and reproduce results from a recent research paper, or simply to use an existing technique that isn’t implemented in MLlib. In this talk, we’ll walk through the process of developing a new machine learning model for Spark. We’ll start with the basics, by considering how we’d design a parallel implementation of a particular unsupervised learning technique. The bulk of the talk will focus on the details you need to know to turn an algorithm design into an efficient parallel implementation on Spark: we’ll start by reviewing a simple RDD-based implementation, show some improvements, point out some pitfalls to avoid, and iteratively extend our implementation to support contemporary Spark features like ML Pipelines and structured query processing. You’ll leave this talk with everything you need to build a new machine learning technique that runs on Spark.
This document contains C++ code examples using numerical methods like Newton-Raphson, successive approximation, and secant methods to find the real roots of various equations. It includes 6 questions/problems with sample code to find roots of equations like x^4 - 11x = 8, e^x - 3x^2 = 0, x^3 - 2x - 3 = 0, and sin(x) + 3cos(x) = 2. For each problem, the code provided the numerical method used, input/output examples, and calculations to iteratively approximate the root to within a set tolerance.
The Ring programming language version 1.3 book - Part 59 of 88Mahmoud Samir Fayed
The document provides details about printing the final intermediate code generated after executing a Ring program.
It begins by showing the output of running a test Ring program and printing the bytecode. This includes the byte code instructions, operation codes, program counter, data, and other details.
It then shows sections of the large intermediate code output, including function and method definitions, variable assignments, calls between functions, and other low-level operations.
The document explains that the output provides a detailed view of the final bytecode generated from the Ring source code after it has been executed by the Ring virtual machine. This allows viewing and understanding the low-level operations performed during program execution.
Super TypeScript II Turbo - FP Remix (NG Conf 2017)Sean May
This talk focuses on typical functional programming paradigms in JavaScript, as implemented in TypeScript.
The goal of this talk was to provide common ground in FP paradigms, between C# .NET developers, Java Spring developers and JS programmers. The slides have been annotated and extended from the talk, to cover intended concepts not explicit in the code examples, themselves.
https://www.youtube.com/watch?v=9oVKjZrgXmU
The web is evolving, we got it. One of the clear consequences is the complexity of our web apps (formerly known as ‘websites’). The conciseness of functional programming and its fundamentals got our attention, but we knew we could do better. And now we have the Reactive programming model, a functional and declarative way of dealing with big amounts of data.
In the center of it we have Observables: objects responsible to keep your application alive, reacting to any mutation your data may have, through any period of time. We’ll take a look on the concepts and also on the lib that implements it in Angular’s core: RxJS. Using the provided operators, we have great power on our hands, doing anything imaginable in a concise, declarative and easy-to-maintain way.
Watch out: observables are here to stay!
The Ring programming language version 1.5.2 book - Part 74 of 181Mahmoud Samir Fayed
This document provides information about low level functions in Ring that allow interaction with C code and pointers. It describes the following functions:
- callgc() forces garbage collection to free temporary variables
- varptr() gets a C pointer for a Ring variable
- space() allocates memory and returns a string pointer
- nullpointer() returns a NULL pointer for optional parameters
- object2pointer() gets a C pointer for a Ring list or object
- pointer2object() converts a C pointer back to a Ring list or object
- ptrcmp() compares two C pointers
It also briefly mentions ringvm functions for getting information about the runtime environment like loaded classes, functions, memory usage and more.
This document defines key probability terms and concepts. It begins by defining a probability experiment, outcomes, sample space, events such as simple, compound and null events. It then discusses union, intersection and complements of events. It also defines equally likely outcomes, mutually exclusive events, exhaustive events, and conditional probability and independence. The document provides examples to illustrate these concepts and definitions. It concludes by discussing approaches to measure probability such as the classical, relative frequency, subjective and axiomatic approaches. It also covers rules of probability including addition, multiplication and conditional probability rules.
We are a growth marketing agency, that helps startups and well establish companies to achieve rapid and sustainable growth. We use various types of marketing and product iterations — content marketing, social media marketing, paid ads, email marketing, SEO and viral strategies, among others, with a purpose to increase the conversion rate and achieve rapid growth of the user base.
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.
1) The document discusses the principle of mathematical induction, which is used to prove that a proposition P(n) is true for all natural numbers n. It involves showing that P(0) is true, and if P(n) is true then P(n+1) is true, which implies P(n) is true for all n.
2) An example uses induction to prove that n < 2n for all positive integers n. It shows the basis step P(1) is true, and inductive step that if P(n) is true then P(n+1) is true.
3) Another example uses induction to prove the summation formula 1 + 2 + ... +
1) The document discusses the principle of mathematical induction, which is used to prove that a proposition P(n) is true for all natural numbers n. It involves showing that P(0) is true, and if P(n) is true then P(n+1) is true, which implies P(n) is true for all n.
2) An example uses induction to prove that n < 2n for all positive integers n. It shows the basis step P(1) is true, and inductive step that if P(n) is true then P(n+1) is true.
3) Another example uses induction to prove the summation formula 1 + 2 + ... +
The simplest and most common form of mathematical induction infers that a statement involving a natural number n (that is, an integer n ≥ 0 or 1) holds for all values of n. The proof consists of two steps:
The base case (or initial case): prove that the statement holds for 0, or 1.
The induction step (or inductive step, or step case): prove that for every n, if the statement holds for n, then it holds for n + 1. In other words, assume that the statement holds for some arbitrary natural number n, and prove that the statement holds for n + 1
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.
1) The document provides four probability problems involving combinations and permutations to further understanding of combinatorics. It gives the solutions and explanations for each problem involving probabilities of card hands.
2) It then introduces conditional probability and uses examples like the Monty Hall problem to illustrate how conditioning on additional information can change a probability. It provides the definition of conditional probability and proves Bayes' theorem.
3) The document discusses independence of events and uses examples like coin flips and natural disasters to demonstrate independence. It also introduces the continuity of probability and uses examples like the Cantor set to illustrate how it allows calculating probabilities of infinite sets.
The simplest and most common form of mathematical induction infers that a statement involving a natural number n (that is, an integer n ≥ 0 or 1) holds for all values of n. The proof consists of two steps:
The base case (or initial case): prove that the statement holds for 0, or 1.
The induction step (or inductive step, or step case): prove that for every n, if the statement holds for n, then it holds for n + 1. In other words, assume that the statement holds for some arbitrary natural number n, and prove that the statement holds for n + 1
This document contains lecture notes on reliability engineering. It covers basic probability theory concepts like probability distributions, random variables, and rules for combining probabilities. It then discusses reliability topics like definitions of reliability, hazard rate, and measures of reliability like mean time to failure. It also covers classifications of engineering systems into series, parallel and other configurations and how to evaluate their reliability. Finally, it discusses discrete and continuous Markov chains and how to model repairable systems using these techniques.
This document discusses mathematical induction, which is a method for proving that a proposition is true for all natural numbers. It involves showing that the proposition is true for the base case, usually 0 or 1, and assuming the proposition is true for some value n to prove it is also true for n+1. Two examples are provided: proving n < 2n for all positive integers n, and proving the formula for summing the first n positive integers. The document also introduces the second principle of mathematical induction, which involves showing the proposition is true for values up to n before proving it for n+1.
This document provides definitions and formulas for key concepts in descriptive statistics, probability, and common probability distributions including:
- Descriptive statistics such as mean, median, mode, variance, and standard deviation.
- Probability concepts such as probability, events, unions/intersections of events, and basic counting rules.
- Common probability distributions like the binomial, uniform, and normal distributions along with their expected values, variances, and probabilities. Formulas for transformations are also included.
The document is intended as a reference sheet for statistics concepts and calculations in a concise format.
Talk at CIRM on Poisson equation and debiasing techniquesPierre Jacob
- The document discusses debiasing techniques for Markov chain Monte Carlo (MCMC) algorithms.
- It introduces the concept of "fishy functions" which are solutions to Poisson's equation and can be used for control variates to reduce bias and variance in MCMC estimators.
- The document outlines different sections including revisiting unbiased estimation through Poisson's equation, asymptotic variance estimation using a novel "fishy function" estimator, and experiments on different examples.
The document summarizes key concepts about the binomial and geometric distributions:
The binomial distribution models the number of successes in a fixed number of yes/no trials where the probability of success is constant. The geometric distribution models the number of trials until the first success. Both have calculators functions and follow patterns for mean, standard deviation, and normal approximations. Formulas for probability mass and cumulative distribution functions are provided.
The document summarizes key concepts about the binomial and geometric distributions:
The binomial distribution models the number of successes in a fixed number of yes/no trials where the probability of success is constant. The geometric distribution models the number of trials until the first success. Both have calculators functions and follow patterns for mean, standard deviation, and normal approximations. Formulas for probability mass and cumulative distribution functions are provided.
Introduction to probability solutions manualKibria Prangon
This summary provides the key information from the document in 3 sentences:
The document summarizes solutions to exercises from the textbook "Introduction to Probability" by Charles M. Grinstead and J. Laurie Snell. The exercises cover topics like coin flips, probability distributions, combinations, and other probability concepts. The solutions involve calculations, proofs, and explanations of probability scenarios to demonstrate understanding of the course material.
The document provides an overview of key concepts in probability theory and stochastic processes. It defines fundamental terms like sample space, events, probability, conditional probability, independence, random variables, and common probability distributions including binomial, Poisson, exponential, uniform, and Gaussian distributions. Examples are given for each concept to illustrate how it applies to modeling random experiments and computing probabilities. The three main axioms of probability are stated. Key properties and formulas for expectation, variance, and conditional expectation are also summarized.
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.
1) Mathematical induction is a method of proof that involves three parts: the base case, inductive hypothesis, and inductive step. It is used to prove statements for all natural numbers.
2) The document provides four examples of proofs by mathematical induction. The first two examples prove formulas for the sums of odd integers and even integers. The third proves an inequality involving factorials.
3) Key aspects of proofs by induction discussed include establishing the base case, assuming the inductive hypothesis is true for an arbitrary value k, and manipulating the k+1 case to substitute the inductive hypothesis and complete the inductive step.
Similar to A/B Testing Theory and Practice (TrueMotion Data Science Lunch Seminar) (20)
Did you know that drowning is a leading cause of unintentional death among young children? According to recent data, children aged 1-4 years are at the highest risk. Let's raise awareness and take steps to prevent these tragic incidents. Supervision, barriers around pools, and learning CPR can make a difference. Stay safe this summer!
We are pleased to share with you the latest VCOSA statistical report on the cotton and yarn industry for the month of March 2024.
Starting from January 2024, the full weekly and monthly reports will only be available for free to VCOSA members. To access the complete weekly report with figures, charts, and detailed analysis of the cotton fiber market in the past week, interested parties are kindly requested to contact VCOSA to subscribe to the newsletter.
Build applications with generative AI on Google CloudMárton Kodok
We will explore Vertex AI - Model Garden powered experiences, we are going to learn more about the integration of these generative AI APIs. We are going to see in action what the Gemini family of generative models are for developers to build and deploy AI-driven applications. Vertex AI includes a suite of foundation models, these are referred to as the PaLM and Gemini family of generative ai models, and they come in different versions. We are going to cover how to use via API to: - execute prompts in text and chat - cover multimodal use cases with image prompts. - finetune and distill to improve knowledge domains - run function calls with foundation models to optimize them for specific tasks. At the end of the session, developers will understand how to innovate with generative AI and develop apps using the generative ai industry trends.
20. Casagrande et al (1978)
Approximate formula gives the desired sample size as a function of , , , and :
where is a "correction factor" given by
with and where denotes the standard normal quantile function, i.e.
is location of the -th quantile for
n p1 p2 α β
n = A
⎡
⎣
⎢
⎢
⎢
1 + 1 +
4( − )p1 p2
A
‾ ‾‾‾‾‾‾‾‾‾‾
√
2( − )p1 p2
⎤
⎦
⎥
⎥
⎥
2
A χ2
A = ,[ + ]z1−α 2 (1 − )p¯ p¯‾ ‾‾‾‾‾‾‾‾√ z1−β (1 − ) + (1 − )p1 p1 p2 p2‾ ‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾√
2
= ( + )/2p¯ p1 p2 zp
= (p)zp Φ−1 p N(0, 1)
21. Example
In [9]: p1, p2 = 0.40, 0.60
alpha = 0.05
beta = 0.05
# Evaluate quantile functions
p_bar = (p1 + p2)/2.0
za = stats.norm.ppf(1 - alpha/2) # Two-sided test
zb = stats.norm.ppf(1 - beta)
# Compute correction factor
A = (za*np.sqrt(2*p_bar*(1-p_bar)) + zb*np.sqrt(p1*(1-p1) + p2*(1-p2)))**2
# Estimate samples required
n = A*(((1 + np.sqrt(1 + 4*(p1-p2)/A))) / (2*(p1-p2)))**2
print n
149.2852619
21
23. In [10]:
So, for test and control combined we'll need at least 1.1 million users.
p1, p2 = 0.0500, 0.0515
alpha = 0.05
beta = 0.05
# Evaluate quantile functions
p_bar = (p1 + p2)/2.0
za = stats.norm.ppf(1 - alpha/2) # Two-sided test
zb = stats.norm.ppf(1 - beta)
# Compute correction factor
A = (za*np.sqrt(2*p_bar*(1-p_bar)) + zb*np.sqrt(p1*(1-p1) + p2*(1-p2)))**2
# Estimate samples required
n = A*(((1 + np.sqrt(1 + 4*(p1-p2)/A))) / (2*(p1-p2)))**2
print n
555118.7638
31
2n =
24. Also, let's verify that this calculation even works...
In [11]: n = 555119
n_trials = 10000
# Simulate experimental results when null is true
control0 = stats.binom.rvs(n, p1, size=n_trials)
test0 = stats.binom.rvs(n, p1, size=n_trials) # Test and control are the sa
me
tables0 = [[[a, n-a], [b, n-b]] for a, b in zip(control0, test0)]
results0 = [stats.chi2_contingency(T) for T in tables0]
decisions0 = [x[1] <= alpha for x in results0]
# Simulate Experimental results when alternate is true
control1 = stats.binom.rvs(n, p1, size=n_trials)
test1 = stats.binom.rvs(n, p2, size=n_trials) # Test and control are differ
ent
tables1 = [[[a, n-a], [b, n-b]] for a, b in zip(control1, test1)]
results1 = [stats.chi2_contingency(T) for T in tables1]
decisions1 = [x[1] <= alpha for x in results1]
# Compute false alarm and correct detection rates
alpha_est = sum(decisions0)/float(n_trials)
power_est = sum(decisions1)/float(n_trials)
print('Theoretical false alarm rate = {:0.4f}, '.format(alpha) +
'empirical false alarm rate = {:0.4f}'.format(alpha_est))
print('Theoretical power = {:0.4f}, '.format(1 - beta) +
'empirical power = {:0.4f}'.format(power_est))
Theoretical false alarm rate = 0.0500, empirical false alarm rate = 0.04
82
Theoretical power = 0.9500, empirical power = 0.9466