probability

MATH 464

HOMEWORK 8

SPRING 2013

The following assignment is to be turned in on Thursday, April 4, 2013.

1. Let X be a Poisson random variable with parameter λ > 0.

a) Find the moment generating function for X.

b) Use your result above to find the mean of the random variable Z = 2X3 − 3X2 + X. c) Consider n ≥ 1, independent, discrete random variables X1, X2, · · · , Xn, and suppose that each are Poisson with parameter λ > 0. Let Z = X1 + X2 + · · · + Xn. Find the pmf of Z.

2. Let X be a negative binomial random variable with parameters n and p. Calculate the variance of X.

3. Let X be an exponential random variable with parameter λ > 0.

a) Let t ≥ 0 and calculate P(X ≥ t). b) Let s,t ≥ 0 and calculate P(X ≥ s + t|X ≥ s). (You can compare your answer to this question with your answer to problem #5 on homework #5.)

4. The gamma function is defined by

Γ(w) =

∫ ∞ 0

xw−1e−x dx

for all w > 0. In terms of this function, a continuous random variable X (with parameters w > 0 and λ > 0) is defined by setting

fX(x) =

{ λw

Γ(w) xw−1e−λx if x > 0,

0 otherwise.

and declaring that X has probability density function fX(x). (fX is called the gamma distribution with parameters w > 0 and λ > 0.)

a) Show that X is a continuous random variable by showing that∫ R fX(t) dt = 1

for all values of w > 0 and λ > 0. 1

2 SPRING 2013

b) Show that for any w > 1,

Γ(w) = (w − 1)Γ(w − 1) Use your result to calculate Γ(n) for any integer n ≥ 2. c) Compute the mean and variance of this random variable X.