Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

3.3. TheΓ,χ^2 ,andβDistributions 185

3.3.20.Determine the constantcso thatf(x)=cx(3−x)^4 , 0 <x<3, zero
elsewhere, is a pdf.

3.3.21. Show that the graph of theβpdf is symmetric about the vertical line
throughx=^12 ifα=β.

3.3.22.Show, fork=1, 2 ,...,n,that

∫ 1

p

n!

(k−1)!(n−k)!

zk−^1 (1−z)n−kdz=

k∑− 1

x=0

(

n

x

)

px(1−p)n−x.

This demonstrates the relationship between the cdfs of theβand binomial distri-
butions.

3.3.23.LetX 1 andX 2 be independent random variables. LetX 1 andY=X 1 +X 2
have chi-square distributions withr 1 andrdegrees of freedom, respectively. Here
r 1 <r. Show thatX 2 has a chi-square distribution withr−r 1 degrees of freedom.
Hint: Write M(t)=E(et(X^1 +X^2 )) and make use of the independence ofX 1 and
X 2.

3.3.24.LetX 1 ,X 2 be two independent random variables having gamma distribu-
tions with parametersα 1 =3,β 1 =3andα 2 =5,β 2 = 1, respectively.
(a)Find the mgf ofY=2X 1 +6X 2.

(b)What is the distribution ofY?

3.3.25.LetXhave an exponential distribution.

(a)Forx>0andy>0, show that

P(X>x+y|X>x)=P(X>y). (3.3.11)

Hence, the exponential distribution has thememorylessproperty. Recall

from Exercise 3.1.9 that the discrete geometric distribution has a similar prop-

erty.

(b)LetF(x) be the cdf of a continuous random variableY. Assume thatF(0) = 0

and 0<F(y)<1fory>0. Suppose property (3.3.11) holds forY. Show

thatFY(y)=1−e−λyfory>0.

Hint:Show thatg(y)=1−FY(y) satisfies the equation

g(y+z)=g(y)g(z),

3.3.26.LetXdenote time until failure of a device and letr(x) denote the hazard
function ofX.

(a)Ifr(x)=cxb;wherecandbare positive constants, show thatXhas aWeibull

distribution; i.e.,

f(x)=

{

cxbexp

{

−cx

b+1

b+1

}

0 <x<∞

0elsewhere.

(3.3.12)