(Introduction to Probability at an Advanced Level) - All Lecture Notes
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Lecture 22: Bivariate Normal Distribution Distribution
6.5 Conditional Distributions General Bivariate Normal Let Z1; Z2 ∼ N (0; 1), which we will use to build a general bivariate normal Lecture 22: Bivariate Normal Distribution distribution. 1 1 2 2 f (z1; z2) = exp − (z1 + z2 ) Statistics 104 2π 2 We want to transform these unit normal distributions to have the follow Colin Rundel arbitrary parameters: µX ; µY ; σX ; σY ; ρ April 11, 2012 X = σX Z1 + µX p 2 Y = σY [ρZ1 + 1 − ρ Z2] + µY Statistics 104 (Colin Rundel) Lecture 22 April 11, 2012 1 / 22 6.5 Conditional Distributions 6.5 Conditional Distributions General Bivariate Normal - Marginals General Bivariate Normal - Cov/Corr First, lets examine the marginal distributions of X and Y , Second, we can find Cov(X ; Y ) and ρ(X ; Y ) Cov(X ; Y ) = E [(X − E(X ))(Y − E(Y ))] X = σX Z1 + µX h p i = E (σ Z + µ − µ )(σ [ρZ + 1 − ρ2Z ] + µ − µ ) = σX N (0; 1) + µX X 1 X X Y 1 2 Y Y 2 h p 2 i = N (µX ; σX ) = E (σX Z1)(σY [ρZ1 + 1 − ρ Z2]) h 2 p 2 i = σX σY E ρZ1 + 1 − ρ Z1Z2 p 2 2 Y = σY [ρZ1 + 1 − ρ Z2] + µY = σX σY ρE[Z1 ] p 2 = σX σY ρ = σY [ρN (0; 1) + 1 − ρ N (0; 1)] + µY = σ [N (0; ρ2) + N (0; 1 − ρ2)] + µ Y Y Cov(X ; Y ) ρ(X ; Y ) = = ρ = σY N (0; 1) + µY σX σY 2 = N (µY ; σY ) Statistics 104 (Colin Rundel) Lecture 22 April 11, 2012 2 / 22 Statistics 104 (Colin Rundel) Lecture 22 April 11, 2012 3 / 22 6.5 Conditional Distributions 6.5 Conditional Distributions General Bivariate Normal - RNG Multivariate Change of Variables Consequently, if we want to generate a Bivariate Normal random variable Let X1;:::; Xn have a continuous joint distribution with pdf f defined of S. -
Effect of Probability Distribution of the Response Variable in Optimal Experimental Design with Applications in Medicine †
mathematics Article Effect of Probability Distribution of the Response Variable in Optimal Experimental Design with Applications in Medicine † Sergio Pozuelo-Campos *,‡ , Víctor Casero-Alonso ‡ and Mariano Amo-Salas ‡ Department of Mathematics, University of Castilla-La Mancha, 13071 Ciudad Real, Spain; [email protected] (V.C.-A.); [email protected] (M.A.-S.) * Correspondence: [email protected] † This paper is an extended version of a published conference paper as a part of the proceedings of the 35th International Workshop on Statistical Modeling (IWSM), Bilbao, Spain, 19–24 July 2020. ‡ These authors contributed equally to this work. Abstract: In optimal experimental design theory it is usually assumed that the response variable follows a normal distribution with constant variance. However, some works assume other probability distributions based on additional information or practitioner’s prior experience. The main goal of this paper is to study the effect, in terms of efficiency, when misspecification in the probability distribution of the response variable occurs. The elemental information matrix, which includes information on the probability distribution of the response variable, provides a generalized Fisher information matrix. This study is performed from a practical perspective, comparing a normal distribution with the Poisson or gamma distribution. First, analytical results are obtained, including results for the linear quadratic model, and these are applied to some real illustrative examples. The nonlinear 4-parameter Hill model is next considered to study the influence of misspecification in a Citation: Pozuelo-Campos, S.; dose-response model. This analysis shows the behavior of the efficiency of the designs obtained in Casero-Alonso, V.; Amo-Salas, M. -
Chapter 6 Continuous Random Variables and Probability
EF 507 QUANTITATIVE METHODS FOR ECONOMICS AND FINANCE FALL 2019 Chapter 6 Continuous Random Variables and Probability Distributions Chap 6-1 Probability Distributions Probability Distributions Ch. 5 Discrete Continuous Ch. 6 Probability Probability Distributions Distributions Binomial Uniform Hypergeometric Normal Poisson Exponential Chap 6-2/62 Continuous Probability Distributions § A continuous random variable is a variable that can assume any value in an interval § thickness of an item § time required to complete a task § temperature of a solution § height in inches § These can potentially take on any value, depending only on the ability to measure accurately. Chap 6-3/62 Cumulative Distribution Function § The cumulative distribution function, F(x), for a continuous random variable X expresses the probability that X does not exceed the value of x F(x) = P(X £ x) § Let a and b be two possible values of X, with a < b. The probability that X lies between a and b is P(a < X < b) = F(b) -F(a) Chap 6-4/62 Probability Density Function The probability density function, f(x), of random variable X has the following properties: 1. f(x) > 0 for all values of x 2. The area under the probability density function f(x) over all values of the random variable X is equal to 1.0 3. The probability that X lies between two values is the area under the density function graph between the two values 4. The cumulative density function F(x0) is the area under the probability density function f(x) from the minimum x value up to x0 x0 f(x ) = f(x)dx 0 ò xm where -
Lecture.7 Poisson Distributions - Properties, Normal Distributions- Properties
Lecture.7 Poisson Distributions - properties, Normal Distributions- properties Theoretical Distributions Theoretical distributions are 1. Binomial distribution Discrete distribution 2. Poisson distribution 3. Normal distribution Continuous distribution Discrete Probability distribution Bernoulli distribution A random variable x takes two values 0 and 1, with probabilities q and p ie., p(x=1) = p and p(x=0)=q, q-1-p is called a Bernoulli variate and is said to be Bernoulli distribution where p and q are probability of success and failure. It was given by Swiss mathematician James Bernoulli (1654-1705) Example • Tossing a coin(head or tail) • Germination of seed(germinate or not) Binomial distribution Binomial distribution was discovered by James Bernoulli (1654-1705). Let a random experiment be performed repeatedly and the occurrence of an event in a trial be called as success and its non-occurrence is failure. Consider a set of n independent trails (n being finite), in which the probability p of success in any trail is constant for each trial. Then q=1-p is the probability of failure in any trail. 1 The probability of x success and consequently n-x failures in n independent trails. But x successes in n trails can occur in ncx ways. Probability for each of these ways is pxqn-x. P(sss…ff…fsf…f)=p(s)p(s)….p(f)p(f)…. = p,p…q,q… = (p,p…p)(q,q…q) (x times) (n-x times) Hence the probability of x success in n trials is given by x n-x ncx p q Definition A random variable x is said to follow binomial distribution if it assumes non- negative values and its probability mass function is given by P(X=x) =p(x) = x n-x ncx p q , x=0,1,2…n q=1-p 0, otherwise The two independent constants n and p in the distribution are known as the parameters of the distribution. -
Extremal Dependence Concepts
Extremal dependence concepts Giovanni Puccetti1 and Ruodu Wang2 1Department of Economics, Management and Quantitative Methods, University of Milano, 20122 Milano, Italy 2Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, ON N2L3G1, Canada Journal version published in Statistical Science, 2015, Vol. 30, No. 4, 485{517 Minor corrections made in May and June 2020 Abstract The probabilistic characterization of the relationship between two or more random variables calls for a notion of dependence. Dependence modeling leads to mathematical and statistical challenges and recent developments in extremal dependence concepts have drawn a lot of attention to probability and its applications in several disciplines. The aim of this paper is to review various concepts of extremal positive and negative dependence, including several recently established results, reconstruct their history, link them to probabilistic optimization problems, and provide a list of open questions in this area. While the concept of extremal positive dependence is agreed upon for random vectors of arbitrary dimensions, various notions of extremal negative dependence arise when more than two random variables are involved. We review existing popular concepts of extremal negative dependence given in literature and introduce a novel notion, which in a general sense includes the existing ones as particular cases. Even if much of the literature on dependence is focused on positive dependence, we show that negative dependence plays an equally important role in the solution of many optimization problems. While the most popular tool used nowadays to model dependence is that of a copula function, in this paper we use the equivalent concept of a set of rearrangements. -
5. the Student T Distribution
Virtual Laboratories > 4. Special Distributions > 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 5. The Student t Distribution In this section we will study a distribution that has special importance in statistics. In particular, this distribution will arise in the study of a standardized version of the sample mean when the underlying distribution is normal. The Probability Density Function Suppose that Z has the standard normal distribution, V has the chi-squared distribution with n degrees of freedom, and that Z and V are independent. Let Z T= √V/n In the following exercise, you will show that T has probability density function given by −(n +1) /2 Γ((n + 1) / 2) t2 f(t)= 1 + , t∈ℝ ( n ) √n π Γ(n / 2) 1. Show that T has the given probability density function by using the following steps. n a. Show first that the conditional distribution of T given V=v is normal with mean 0 a nd variance v . b. Use (a) to find the joint probability density function of (T,V). c. Integrate the joint probability density function in (b) with respect to v to find the probability density function of T. The distribution of T is known as the Student t distribution with n degree of freedom. The distribution is well defined for any n > 0, but in practice, only positive integer values of n are of interest. This distribution was first studied by William Gosset, who published under the pseudonym Student. In addition to supplying the proof, Exercise 1 provides a good way of thinking of the t distribution: the t distribution arises when the variance of a mean 0 normal distribution is randomized in a certain way. -
Laws of Total Expectation and Total Variance
Laws of Total Expectation and Total Variance Definition of conditional density. Assume and arbitrary random variable X with density fX . Take an event A with P (A) > 0. Then the conditional density fXjA is defined as follows: 8 < f(x) 2 P (A) x A fXjA(x) = : 0 x2 = A Note that the support of fXjA is supported only in A. Definition of conditional expectation conditioned on an event. Z Z 1 E(h(X)jA) = h(x) fXjA(x) dx = h(x) fX (x) dx A P (A) A Example. For the random variable X with density function 8 < 1 3 64 x 0 < x < 4 f(x) = : 0 otherwise calculate E(X2 j X ≥ 1). Solution. Step 1. Z h i 4 1 1 1 x=4 255 P (X ≥ 1) = x3 dx = x4 = 1 64 256 4 x=1 256 Step 2. Z Z ( ) 1 256 4 1 E(X2 j X ≥ 1) = x2 f(x) dx = x2 x3 dx P (X ≥ 1) fx≥1g 255 1 64 Z ( ) ( )( ) h i 256 4 1 256 1 1 x=4 8192 = x5 dx = x6 = 255 1 64 255 64 6 x=1 765 Definition of conditional variance conditioned on an event. Var(XjA) = E(X2jA) − E(XjA)2 1 Example. For the previous example , calculate the conditional variance Var(XjX ≥ 1) Solution. We already calculated E(X2 j X ≥ 1). We only need to calculate E(X j X ≥ 1). Z Z Z ( ) 1 256 4 1 E(X j X ≥ 1) = x f(x) dx = x x3 dx P (X ≥ 1) fx≥1g 255 1 64 Z ( ) ( )( ) h i 256 4 1 256 1 1 x=4 4096 = x4 dx = x5 = 255 1 64 255 64 5 x=1 1275 Finally: ( ) 8192 4096 2 630784 Var(XjX ≥ 1) = E(X2jX ≥ 1) − E(XjX ≥ 1)2 = − = 765 1275 1625625 Definition of conditional expectation conditioned on a random variable. -
A Study of Non-Central Skew T Distributions and Their Applications in Data Analysis and Change Point Detection
A STUDY OF NON-CENTRAL SKEW T DISTRIBUTIONS AND THEIR APPLICATIONS IN DATA ANALYSIS AND CHANGE POINT DETECTION Abeer M. Hasan A Dissertation Submitted to the Graduate College of Bowling Green State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY August 2013 Committee: Arjun K. Gupta, Co-advisor Wei Ning, Advisor Mark Earley, Graduate Faculty Representative Junfeng Shang. Copyright c August 2013 Abeer M. Hasan All rights reserved iii ABSTRACT Arjun K. Gupta, Co-advisor Wei Ning, Advisor Over the past three decades there has been a growing interest in searching for distribution families that are suitable to analyze skewed data with excess kurtosis. The search started by numerous papers on the skew normal distribution. Multivariate t distributions started to catch attention shortly after the development of the multivariate skew normal distribution. Many researchers proposed alternative methods to generalize the univariate t distribution to the multivariate case. Recently, skew t distribution started to become popular in research. Skew t distributions provide more flexibility and better ability to accommodate long-tailed data than skew normal distributions. In this dissertation, a new non-central skew t distribution is studied and its theoretical properties are explored. Applications of the proposed non-central skew t distribution in data analysis and model comparisons are studied. An extension of our distribution to the multivariate case is presented and properties of the multivariate non-central skew t distri- bution are discussed. We also discuss the distribution of quadratic forms of the non-central skew t distribution. In the last chapter, the change point problem of the non-central skew t distribution is discussed under different settings. -
The Open Handbook of Formal Epistemology
THEOPENHANDBOOKOFFORMALEPISTEMOLOGY Richard Pettigrew &Jonathan Weisberg,Eds. THEOPENHANDBOOKOFFORMAL EPISTEMOLOGY Richard Pettigrew &Jonathan Weisberg,Eds. Published open access by PhilPapers, 2019 All entries copyright © their respective authors and licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License. LISTOFCONTRIBUTORS R. A. Briggs Stanford University Michael Caie University of Toronto Kenny Easwaran Texas A&M University Konstantin Genin University of Toronto Franz Huber University of Toronto Jason Konek University of Bristol Hanti Lin University of California, Davis Anna Mahtani London School of Economics Johanna Thoma London School of Economics Michael G. Titelbaum University of Wisconsin, Madison Sylvia Wenmackers Katholieke Universiteit Leuven iii For our teachers Overall, and ultimately, mathematical methods are necessary for philosophical progress. — Hannes Leitgeb There is no mathematical substitute for philosophy. — Saul Kripke PREFACE In formal epistemology, we use mathematical methods to explore the questions of epistemology and rational choice. What can we know? What should we believe and how strongly? How should we act based on our beliefs and values? We begin by modelling phenomena like knowledge, belief, and desire using mathematical machinery, just as a biologist might model the fluc- tuations of a pair of competing populations, or a physicist might model the turbulence of a fluid passing through a small aperture. Then, we ex- plore, discover, and justify the laws governing those phenomena, using the precision that mathematical machinery affords. For example, we might represent a person by the strengths of their beliefs, and we might measure these using real numbers, which we call credences. Having done this, we might ask what the norms are that govern that person when we represent them in that way. -
Discrete Probability Distributions Uniform Distribution Bernoulli
Discrete Probability Distributions Uniform Distribution Experiment obeys: all outcomes equally probable Random variable: outcome Probability distribution: if k is the number of possible outcomes, 1 if x is a possible outcome p(x)= k ( 0 otherwise Example: tossing a fair die (k = 6) Bernoulli Distribution Experiment obeys: (1) a single trial with two possible outcomes (success and failure) (2) P trial is successful = p Random variable: number of successful trials (zero or one) Probability distribution: p(x)= px(1 − p)n−x Mean and variance: µ = p, σ2 = p(1 − p) Example: tossing a fair coin once Binomial Distribution Experiment obeys: (1) n repeated trials (2) each trial has two possible outcomes (success and failure) (3) P ith trial is successful = p for all i (4) the trials are independent Random variable: number of successful trials n x n−x Probability distribution: b(x; n,p)= x p (1 − p) Mean and variance: µ = np, σ2 = np(1 − p) Example: tossing a fair coin n times Approximations: (1) b(x; n,p) ≈ p(x; λ = pn) if p ≪ 1, x ≪ n (Poisson approximation) (2) b(x; n,p) ≈ n(x; µ = pn,σ = np(1 − p) ) if np ≫ 1, n(1 − p) ≫ 1 (Normal approximation) p Geometric Distribution Experiment obeys: (1) indeterminate number of repeated trials (2) each trial has two possible outcomes (success and failure) (3) P ith trial is successful = p for all i (4) the trials are independent Random variable: trial number of first successful trial Probability distribution: p(x)= p(1 − p)x−1 1 2 1−p Mean and variance: µ = p , σ = p2 Example: repeated attempts to start -
Probability Cheatsheet V2.0 Thinking Conditionally Law of Total Probability (LOTP)
Probability Cheatsheet v2.0 Thinking Conditionally Law of Total Probability (LOTP) Let B1;B2;B3; :::Bn be a partition of the sample space (i.e., they are Compiled by William Chen (http://wzchen.com) and Joe Blitzstein, Independence disjoint and their union is the entire sample space). with contributions from Sebastian Chiu, Yuan Jiang, Yuqi Hou, and Independent Events A and B are independent if knowing whether P (A) = P (AjB )P (B ) + P (AjB )P (B ) + ··· + P (AjB )P (B ) Jessy Hwang. Material based on Joe Blitzstein's (@stat110) lectures 1 1 2 2 n n A occurred gives no information about whether B occurred. More (http://stat110.net) and Blitzstein/Hwang's Introduction to P (A) = P (A \ B1) + P (A \ B2) + ··· + P (A \ Bn) formally, A and B (which have nonzero probability) are independent if Probability textbook (http://bit.ly/introprobability). Licensed and only if one of the following equivalent statements holds: For LOTP with extra conditioning, just add in another event C! under CC BY-NC-SA 4.0. Please share comments, suggestions, and errors at http://github.com/wzchen/probability_cheatsheet. P (A \ B) = P (A)P (B) P (AjC) = P (AjB1;C)P (B1jC) + ··· + P (AjBn;C)P (BnjC) P (AjB) = P (A) P (AjC) = P (A \ B1jC) + P (A \ B2jC) + ··· + P (A \ BnjC) P (BjA) = P (B) Last Updated September 4, 2015 Special case of LOTP with B and Bc as partition: Conditional Independence A and B are conditionally independent P (A) = P (AjB)P (B) + P (AjBc)P (Bc) given C if P (A \ BjC) = P (AjC)P (BjC). -
Chapter 5 Sections
Chapter 5 Chapter 5 sections Discrete univariate distributions: 5.2 Bernoulli and Binomial distributions Just skim 5.3 Hypergeometric distributions 5.4 Poisson distributions Just skim 5.5 Negative Binomial distributions Continuous univariate distributions: 5.6 Normal distributions 5.7 Gamma distributions Just skim 5.8 Beta distributions Multivariate distributions Just skim 5.9 Multinomial distributions 5.10 Bivariate normal distributions 1 / 43 Chapter 5 5.1 Introduction Families of distributions How: Parameter and Parameter space pf /pdf and cdf - new notation: f (xj parameters ) Mean, variance and the m.g.f. (t) Features, connections to other distributions, approximation Reasoning behind a distribution Why: Natural justification for certain experiments A model for the uncertainty in an experiment All models are wrong, but some are useful – George Box 2 / 43 Chapter 5 5.2 Bernoulli and Binomial distributions Bernoulli distributions Def: Bernoulli distributions – Bernoulli(p) A r.v. X has the Bernoulli distribution with parameter p if P(X = 1) = p and P(X = 0) = 1 − p. The pf of X is px (1 − p)1−x for x = 0; 1 f (xjp) = 0 otherwise Parameter space: p 2 [0; 1] In an experiment with only two possible outcomes, “success” and “failure”, let X = number successes. Then X ∼ Bernoulli(p) where p is the probability of success. E(X) = p, Var(X) = p(1 − p) and (t) = E(etX ) = pet + (1 − p) 8 < 0 for x < 0 The cdf is F(xjp) = 1 − p for 0 ≤ x < 1 : 1 for x ≥ 1 3 / 43 Chapter 5 5.2 Bernoulli and Binomial distributions Binomial distributions Def: Binomial distributions – Binomial(n; p) A r.v.