probability

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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.