Report

Functions of Random Variables Notes of STAT 6205 by Dr. Fan Overview • Chapter 5 • Functions of One random variable o General: distribution function approach o Change-of-variable approach • Functions of Two random variables o Change-of-variable approach • • • • Functions of Independent random variables Order statistics The Moment Generating Function approach Random functions associated with normal distributions o Student’s t-distribution • The Central Limit Theorem o Normal approximation of binomial distribution • (Section 10.5) Chebyshev’s Inequality and convergence in probability 6205-Ch5 2 General Method: Distribution Function Approach • Goal: to find the distribution of Y=h(X) • When: the pdf of X, f(x) is known • Then the cdf of Y, G(y) is: G ( y ) P [Y y ] P [ h ( X ) y ] f ( x ) dx h ( x ) y • And the pdf of Y, g(y)=G’(y) 6205-Ch5 3 Examples/Exercises • Let X~U(0,10) and Y=X^3. Find the cdf and pdf of Y • Let X~Exp(mu=2) and Y=Exp(X). Find the cdf and pdf of Y • Let X~Gamma(a,b) and X=log(Y). Find the pdf of Y (Loggamma distribution) 6205-Ch5 4 Change of Variable Approach • When: the pdf of X is known and Y=h(X), a monotonic function (i.e. its inverse function exists; X = V(Y) ) 6205-Ch5 5 Examples/Exercises Let Y=(1-X)^3 and find its pdf g(y) • Problem 1: f(x)=x/2, 0<x<2 • Problem 2: f(x)=3(1-x)^2, 0<x<1 • Problem 3: verify that the g attained in problem 2 is a proper pdf • Problem4: revisit the problems in Slide 4 6205-Ch5 6 Transformations of Two Random Variables • Let f(x1,x2) be the joint pdf of X1,X2 • Let Y1=u1(X1,X2) and Y2=u2(X1,X2) • where u1, u2 have inverse functions, that is, X1=v1(Y1,Y2) and X2=v2(Y1,Y2) • Goal: find the joint pdf of Y1,Y2, g(y1,y2) g ( y 1 , y 2 ) | J | f [ v1 ( y 1 , y 2 ), v 2 ( y 1 , y 2 )], where ( y 1 , y 2 ) S 1 x 1 y1 and the Jacobian J x 2 y1 6205-Ch5 x 1 y 2 . x 2 y 2 7 Examples/Exercises 1. f(x1,x2)=2 where 0<x1<x2<1; Y1=X1/X2 and Y2=X2 2. X1, X2 are independent exp(1) variables; Y1=X1-X2 and Y2=X1+X2 3. Reading: Examples 5.2-3, 4 6205-Ch5 8 Independent Random Variables • Let X1, X2, …,Xn be independent random variables • Joint pmf (or pdf) of X1, X2, …, Xn: f(x1,x2,…,xn)=f1(x1)f2(x2)…fn(xn) • Random sample from a distribution f(x): X1, X2, … Xn are independent and identically distributed; f(x1,x2,…,xn)=f(x1)f(x2)…f(xn) 6205-Ch5 9 Examples/Exercises • Let X1, X2, …, Xn be a random sample from Exp(0.5). Find the joint p.d.f of this sample. • Exercise: What is the probability of seeing at least one Xi less than one? Exactly one less than one? 6205-Ch5 10 Functions of Independent R. V.s Theorem 5.3-2 Let X1, X2, …, Xn be independent r. v.s. Then: E [ u 1 ( X 1 ) u 2 ( X 2 )... u n ( X n )] Theorem 5.3-3 (page 238) 6205-Ch5 E [u i ( X i )] i 11 Examples/Exercises • Given a random sample of size n from a distribution with mean mu and SD sigma, find the mean and variance of the sample mean 6205-Ch5 12 Moment Generating Function 6205-Ch5 13 Examples/Exercises • Example: Prove that the sum of i.i.d. Ber(p) r.v.s is a Bin(n, p) r. v. • Exercise: Prove that the sum of i.i.d. Exp(mu) r. v.s is a Gamma(a=n, b=0.5) r. v. 1) What is the m.g.f. of Exp(mu)? 2) What is the m.g.f. of Gamma(a,b)? 3) Prove this problem using m.g.f. 6205-Ch5 14 Random Variables Assoc. With Normal Distributions Theorem 1: The distribution of the sum of i.i.d. normal r. v.s is also normal Theorem 2: The distribution of the sum of normal r. v.s is also normal Theorem 3: The distribution of the average of normal r. v.s is also normal 6205-Ch5 15 Student’s t-distribution 6205-Ch5 16 Proof: 1) Show S^2 and X-bar are independent 2) Use m.g.f to prove the distribution is chi-square Example: Show that the one-sample t test statistic is tdistributed with (n-1) degree of freedom T X S/ 6205-Ch5 ~ t ( n 1) n 17 Features of t distribution t(r) • Shape: Bell-shaped • Center and Spread: mean=0 if r > 1 variance =r/(r-2) if r > 2 (undefined otherwise) • M.G.F. does not exist • Asymptotic distribution: (show simulation results) As d.f. r goes to infinity, t(r) approaches to N(0,1) 6205-Ch5 18 Central Limit Theorem 6205-Ch5 19 Examples/Exercises • Illustration: Bin(n, p) goes to Normal as n goes to infinity [Aplia: STAT 1000 homework 4 Q3] • Problem: Let X-bar be the mean of a random sample of n=25 currents in a strip of wire in which each measurement has a mean of 15 and a variance of 4. Estimate the probability of X-bar falling between 14.4 and 15.6. • Problem: Suppose BART wants to perform some quality control. They know the waiting time for one at a BART station is U(10,30). In a random sample of 30 people, what tis the (approximate) probability that the average waiting time is more than 22 minutes? Recall the mean and variance for U(10,30) is 20 and 33.33 respectively. 6205-Ch5 20 Chebyshev’s Inequality If the r. v. X has a mean m and variance s^2, then for every k > 1, P (| X | k ) 1 / k 2 Q: how to use this inequality to set up a lower bound of P(|X - m|< ks)? Example: Use this inequality to find a lower bound of the probability that X is no more than 2 S.D. from the mean. Is the lower bound close to the exact probability if X ~ N( m, s^2 ) 6205-Ch5 21 Example: Tossing a Coin If we want to estimate p, the chance of heads for a given coin, how many times share we toss it in order to get a sufficient accurate estimate? Let Y be the # of heads on n flips; sample estimate of p, p-hat = Y/n. Use the Chebyshev’s Inequality to find the required sample size n. 6205-Ch5 22 (Weak) Law of Large Number Let X1, X2, …, Xn be i.i.d. r. v.s with finite mean m and finite S.D. s. Then X-bar converges to m in probability. Proof. By Chebychev’s Inequality. 6205-Ch5 23