probability7

Math 464 - Fall 13 - Homework 8

1. X and Y are independent random variables, each of which has the stan- dard normal distribution. Show that Z = X/Y has a Cauchy distribution.

2. Let X be a standard normal random variable, and let Y = σX + µ where σ > 0. (a) Show that the pdf of Y is normal with mean µ and variance σ2. (b) Show that moment generating function of X is exp(t2/2). (c) Show that the mgf of Y is given by the formula on the formula sheet. (Hint: recall the proposition from class about the mgf of aX +b. This should take almost no computation.)

3. (Exposition) In class we stated a theorem that says that if X and Y are independent continuous random variables and g and h are functions from R to R, then g(X) and h(Y ) are independent random variables. We only proved it for the special case that g and h are increasing functions. In this problem you prove for two more special cases. (a) Prove that if X and Y are independent then X2 and Y 2 are independent. (b) Prove that if X and Y are independent then X and −Y are independent.

4. The Laplace distribution is

f(x) = 1

2 λe−λ|x|, −∞ < x < ∞

where λ > 0 is a parameter. Compute the moment generating function and use it to find the mean and variance.

5. Let X and Y be independent random variables. They each have the exponential distribution with the same λ. Let Z = Y − X. The goal of this problem is to find the density of Z using moment generating functions. (There should be very little computation in your solution.) (a) Find the mgf of −X. Hint: think of −X as (−1)X and recall the propo- sition from class about the mgf of aX + b. (b) Use the fact that −X and Y are independent (which you proved in a previous problem) to find the mgf of Z. (c) Find the density of Z. Hint: don’t compute - find a RV with the same moment generating function.

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