Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

164 Some Special Distributions

x<-0:15; par(mfrow=c(3,3)); p <- 1:9/10

for(j in p){plot(dbinom(x,15,j)~x); title(paste("p=",j))}

(c)LetY =Xn,whereXhas ab(n, 0 .05) distribution. Obtain the plots of the

pmfs ofYforn=10, 20 , 50 ,200. Comment on the plots (what do the plots

seem to be converging to asngets large?).

3.1.9.Ifx=ris the unique mode of a distribution that isb(n, p), show that

(n+1)p− 1 <r<(n+1)p.

This substantiates the comments made in Part (b) of Exercise 3.1.8.

Hint: Determine the values ofxfor whichp(x+1)/p(x)>1.

3.1.10.SupposeXisb(n, p). Then by definition the pmf is symmetric if and only
ifp(x)=p(n−x), forx=0,...,n. Show that the pmf is symmetric if and only if
p=1/2.

3.1.11.Toss two nickels and three dimes at random. Make appropriate assumptions
and compute the probability that there are more heads showing on the nickels than
on the dimes.

3.1.12.LetX 1 ,X 2 ,...,Xk− 1 have a multinomial distribution.

(a)Find the mgf ofX 2 ,X 3 ,...,Xk− 1.

(b)What is the pmf ofX 2 ,X 3 ,...,Xk− 1?

(c)Determine the conditional pmf ofX 1 given thatX 2 =x 2 ,...,Xk− 1 =xk− 1.

(d)What is the conditional expectationE(X 1 |x 2 ,...,xk− 1 )?

3.1.13.LetXbeb(2,p)andletYbeb(4,p). IfP(X≥1) =^59 , findP(Y≥1).

3.1.14.LetXhave a binomial distribution with parametersnandp=^13. Deter-
mine the smallest integerncanbesuchthatP(X≥1)≥ 0 .85.

3.1.15.LetXhave the pmfp(x)=(^13 )(^23 )x,x=0, 1 , 2 , 3 ,..., zero elsewhere. Find
the conditional pmf ofXgiven thatX≥3.

3.1.16.One of the numbers 1, 2 ,...,6 is to be chosen by casting an unbiased die.

Let this random experiment be repeated five independent times. Let the random

variableX 1 be the number of terminations in the set{x:x=1, 2 , 3 }and let

the random variableX 2 be the number of terminations in the set{x:x=4, 5 }.

ComputeP(X 1 =2,X 2 =1).

3.1.17.Show that the moment generating function of the negative binomial dis-
tribution isM(t)=pr[1−(1−p)et]−r. Find the mean and the variance of this
distribution.
Hint: In the summation representingM(t), make use of the negative binomial
series.^1

(^1) See, for example,Mathematical Commentsreferenced in the Preface.