prob7

 typed answer required.

 

strict due.

 

you should check questions first and let me know if you can complete them 

Math 464 – Fall

1

3 – 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.

1