Functions of Random Variables Notes of STAT 6205 by Dr. Fan

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1 Functions of Random Variables Notes of STAT 605 by Dr. Fan Overview Chapter 5 Functions of One random variable o o General: distribution function approach 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 0.5) Chebyshev s Inequality and convergence in probability 605-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] = h( x) y f ( x) dx And the pdf of Y, g(y)=g (y) 605-Ch5 3 Let X~U(0,0) and Y=X^3. Find the cdf and pdf of Y Let X~Exp(mu=) 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) 605-Ch5 4

3 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) ) 605-Ch5 5 Let Y=(-X)^3 and find its pdf g(y) Problem : f(x)=x/, 0<x< Problem : f(x)=3(-x)^, 0<x< Problem 3: verify that the g attained in problem is a proper pdf Problem4: revisit the problems in Slide Ch5 6

4 Transformations of Two Random Variables Let f(x,x) be the joint pdf of X,X Let Y=u(X,X) and Y=u(X,X) where u, u have inverse functions, that is, X=v(Y,Y) and X=v(Y,Y) Goal: find the joint pdf of Y,Y, g(y,y) g( y, y ) = J f [ v ( y, y ), v ( y, y )], where ( y, y ) S and the Jacobian J = x y x y x y x y. 605-Ch5 7. f(x,x)= where 0<x<x<; Y=X/X and Y=X. X, X are independent exp() variables; Y=X-X and Y=X+X 3. Reading: Examples 5.-3, Ch5 8

5 Independent Random Variables Let X, X,,Xn be independent random variables Joint pmf (or pdf) of X, X,, Xn: f(x,x,,xn)=f(x)f(x) fn(xn) Random sample from a distribution f(x): X, X, Xn are independent and identically distributed; f(x,x,,xn)=f(x)f(x) f(xn) 605-Ch5 9 Let X, X,, 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? 605-Ch5 0

6 Functions of Independent R. V.s Theorem 5.3- Let X, X,, Xn be independent r. v.s. Then: E [ u( X) u( X )... un( X n)] = E[ ui ( X i )] i Theorem (page 38) 605-Ch5 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 605-Ch5

7 Moment Generating Function 605-Ch5 3 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. ) What is the m.g.f. of Exp(mu)? ) What is the m.g.f. of Gamma(a,b)? 3) Prove this problem using m.g.f. 605-Ch5 4

8 Random Variables Assoc. With Normal Distributions Theorem : The distribution of the sum of i.i.d. normal r. v.s is also normal Theorem : 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 605-Ch5 5 Student s s t-distributiont 605-Ch5 6

9 Proof: )Show S^ and X-bar are independent )Use m.g.f to prove the distribution is chi-square Example: Show that the one-sample t test statistic is t- distributed with (n-) degree of freedom T X µ = ~ t( n ) S / n 605-Ch5 7 Features of t distribution t(r) Shape: Bell-shaped Center and Spread: mean=0 if r > variance =r/(r-) if r > (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,) 605-Ch5 8

10 Central Limit Theorem 605-Ch5 9 Illustration: Bin(n, p) goes to Normal as n goes to infinity [Aplia: STAT 000 homework 4 Q3] Problem: Let X-bar be the mean of a random sample of n=5 currents in a strip of wire in which each measurement has a mean of 5 and a variance of 4. Estimate the probability of X-bar falling between 4.4 and 5.6. Problem: Suppose BART wants to perform some quality control. They know the waiting time for one at a BART station is U(0,30). In a random sample of 30 people, what tis the (approximate) probability that the average waiting time is more than minutes? Recall the mean and variance for U(0,30) is 0 and respectively. 605-Ch5 0

11 Chebyshev s s Inequality If the r. v. X has a mean µ and variance σ^, then for every k >, P( X µ kσ ) / k Q: how to use this inequality to set up a lower bound of P( X - µ < κσ)? Example: Use this inequality to find a lower bound of the probability that X is no more than S.D. from the mean. Is the lower bound close to the exact probability if X ~ N( µ, σ^ ) 605-Ch5 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. 605-Ch5

12 (Weak) Law of Large Number Let X, X,, Xn be i.i.d. r. v.s with finite mean µ and finite S.D. σ. Then X-bar converges to µ in probability. Proof. By Chebychev s Inequality. 605-Ch5 3