Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

3.3. TheΓ,χ^2 ,andβDistributions 173

3.2.14.LetXhave a Poisson distribution. IfP(X=1)=P(X= 3), find the

mode of the distribution.

3.2.15.LetXhave a Poisson distribution with mean 1. Compute, if it exists, the

expected valueE(X!).

3.2.16.LetXandYhave the joint pmfp(x, y)=e−^2 /[x!(y−x)!],y=0, 1 , 2 ,...,

x=0, 1 ,...,y, zero elsewhere.

(a)Find the mgfM(t 1 ,t 2 ) of this joint distribution.

(b)Compute the means, the variances, and the correlation coefficient ofXand

Y.

(c)Determine the conditional meanE(X|y).

Hint: Note that

∑y

x=0

[exp(t 1 x)]y!/[x!(y−x)!] = [1 + exp(t 1 )]y.

Why?

3.2.17.LetX 1 andX 2 be two independent random variables. Suppose thatX 1 and
Y =X 1 +X 2 have Poisson distributions with meansμ 1 andμ>μ 1 , respectively.
Find the distribution ofX 2.

3.3 The Γ,χ^2 ,andβDistributions

In this section we introduce the continuous gamma Γ-distribution and several as-
sociated distributions. The support for the Γ-distribution is the set of positive real
numbers. This distribution and its associated distributions are rich in applications
in all areas of science and business. These applications include their use in modeling
lifetimes, failure times, service times, and waiting times.
The definition of the Γ-distribution requires the Γ function from calculus. It is
proved in calculus that the integral
∫∞

0

yα−^1 e−ydy

exists forα>0 and that the value of the integral is a positive number. The integral
is called thegamma functionofα,andwewrite

Γ(α)=

∫∞

0

yα−^1 e−ydy.

Ifα= 1, clearly

Γ(1) =

∫∞

0

e−ydy=1.