Chapter 3 sections. SKIP: 3.10 Markov Chains. SKIP: pages Chapter 3 - continued

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1 Chapter 3 sections Chapter 3 - continued 3.1 Random Variables and Discrete Distributions 3.2 Continuous Distributions 3.3 The Cumulative Distribution Function 3.4 Bivariate Distributions 3.5 Marginal Distributions 3.6 Conditional Distributions Just skim: 3.7 Multivariate Distributions (generalization of bivariate) 3.9 Functions of Two or More Random Variables SKIP: pages SKIP: 3.10 Markov Chains STA 611 (Lecture 05) Random Variables and Distributions 1 / 13

2 Transformations of Random Variables If X is a random variable then any function of X, g(x), is also a random variable Sometimes we are interested in Y = g(x) and need the distribution of Y Example: Say we have the distribution of the service rate X, then what is the distribution of the average waiting time Y = 1/X? We can use the distribution of X to get the distribution of Y : P(Y A) = P(g(X) A) Depending on the function g we can sometimes obtain a tractable expression for the probability of Y Example: P(Y y) = P(1/X y) = P(X 1/y) STA 611 (Lecture 05) Random Variables and Distributions 2 / 13

3 Transformations of Random Variables Inverse mapping Let g(x) : X Y. The inverse mapping is defined as g 1 (A) = {x X : g(x) A} For a set of one point we write g 1 ({y}) = g 1 (y) = {x X : g(x) = y} We can therefore write: P(Y A) = P(g(X) A) = P({x X : g(x) A}) = P(X g 1 (A)) STA 611 (Lecture 05) Random Variables and Distributions 3 / 13

4 Transformation of a discrete random variable If Y = g(x) where X is a discrete r.v. with support X then Y is also a discrete r.v. and f Y (y) = P(Y = y) = P(X g 1 (y)) = P(X = x) = f X (x) x g 1 (y) x g 1 (y) for all y Y = {y : y = g(x), x X } Example: Let X Binomial(n, p), i.e. ( ) n f X (x) = P(X = x) = p x (1 p) n x, x x = 0, 1, 2,..., n What is the pf of Y = n X (i.e. the number of failures)? STA 611 (Lecture 05) Random Variables and Distributions 4 / 13

5 Example 1: Exponential distribution (Continuous r.v.) Let X be a random variable with pdf f X (x) = λ exp( λx), x > 0 where λ is a positive constant We say that X is exponentially distributed with parameter (rate) λ or X Exp(λ). 1 What is the cdf of X? 2 What is the distribution of Y = αx where α is a positive constant? 3 What is the distribution of W = X 2? STA 611 (Lecture 05) Random Variables and Distributions 5 / 13

6 Example 2: Double exponential distribution Also called Laplace distribution Let X be a random variable with pdf f X (x) = λ 2 exp( λ x ), x R where λ is a positive constant What is the cdf of X? What is the cdf of W = X 2? STA 611 (Lecture 05) Random Variables and Distributions 6 / 13

7 Example 3: cdf transformation Again we consider the exponential distribution. Let X Exp(λ), then X has the pdf where λ is a positive constant. f X (x) = λe λx, x > 0 (a) Let F X (x) be the cdf found in Ex. 1. Find the distribution of Y = F X (X). (b) Find the inverse cdf F 1 1 X and the distribution of FX (U) where U Uniform(0, 1). STA 611 (Lecture 05) Random Variables and Distributions 7 / 13

8 Probability integral transformation Theorem 1 Let X have a continuous cdf F and let Y = F(X). Then F(X) Uniform(0, 1). 2 Let Y Uniform(0, 1) and let F be a continuous cdf with quantile function F 1. Then X = F 1 (Y ) has cdf F. This theorem is useful when we want to generate random numbers from some distribution. If F 1 is available in closed form we can simply generate uniform random numbers and then transform them using F 1. Therefore, much of the effort concerning generating (pseudo) random numbers has been concentrated on generating uniform random numbers. STA 611 (Lecture 05) Random Variables and Distributions 8 / 13

9 Monotone transformations of continuous r.v. s Theorem Let X be a random variable with pdf f X (x) and support X and let Y = g(x) where g is a monotone function. Suppose f X (x) is continuous on X and that g 1 (y) has a continuous derivative on Y = {y : y = g(x), x X }. Then the pdf of Y is { f f Y (y) = X (g 1 (y)) d dy g 1 (y) if y Y 0 otherwise Refer to Theorem for the multivariate case. STA 611 (Lecture 05) Random Variables and Distributions 9 / 13

10 Example 4: Transformation of the Gamma distribution Consider the Gamma distribution with parameters n and β Let X Gamma(n, β). Then X has the pdf f (x) = What is the pdf of Y = 1/X? 1 (n 1)!β n x n 1 e x/β The distribution of Y is called the Inverse gamma distribution. STA 611 (Lecture 05) Random Variables and Distributions 10 / 13

11 Linear functions Chapter 3 - continued A straightforward corollary: Linear function Let X be a random variable with pdf f X (x) and let Y = ax + b, a 0. Then f Y (y) = 1 ( ) y b a f a Example: Let X have the pdf f X (x) = 1 σ 2π exp ( ) (x µ)2 2σ 2 This is the pdf of the Normal distribution with parameters µ (mean) and σ 2 (variance). Notation: X N(µ, σ 2 ). Find the pdf of Y = X µ σ. STA 611 (Lecture 05) Random Variables and Distributions 11 / 13

12 3.9 Functions of Two or More Random Variables Sum of two random variables Convolution What is the distribution of Z = X + Y? If X and Y discrete random variables we get P(Z = z) = i P(X = i, Y = z i) P(Z = z) = i P(X = i)p(y = z i) if X and Y are independent X and Y continuous random variables: f Z (z) = f Z (z) = f X,Y (t, z t)dt f X (t)f Y (z t)dt this is called the convolution formula. if X and Y are independent Example: What is the distribution of X + Y for independent standard normals X and Y? STA 611 (Lecture 05) Random Variables and Distributions 12 / 13

13 END OF CHAPTER 3 STA 611 (Lecture 05) Random Variables and Distributions 13 / 13