Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

3.1. The Binomial and Related Distributions 163

EXERCISES

3.1.1.If the mgf of a random variableXis (^13 +^23 et)^5 , findP(X= 2 or 3). Verify
using the R functiondbinom.

3.1.2.The mgf of a random variableXis (^23 +^13 et)^9.

(a)Show that

P(μ− 2 σ<X<μ+2σ)=

∑^5

x=1

(

9

x

)(

1

3

)x(

2

3

) 9 −x

.

(b)Use R to compute the probability in Part (a).

3.1.3.IfXisb(n, p), show that

E

(

X

n

)

=p and E

[(

X

n

−p

) 2 ]

=

p(1−p)

n

.

3.1.4.Let the independent random variablesX 1 ,X 2 ,...,X 40 be iid with the com-
mon pdff(x)=3x^2 , 0 <x<1, zero elsewhere. Find the probability that at least
35 of theXi’s exceed^12.

3.1.5.Over the years, the percentage of candidates passing an entrance exam to a

prestigious law school is 20%. At one of the testing centers, a group of 50 candidates

take the exam and 20 pass. Is this odd? Answer on the basis thatX≥20 where

Xis the number that pass in a group of 50 when the probability of a pass is 0.2.

3.1.6.LetY be the number of successes throughoutnindependent repetitions of

a random experiment with probability of successp=^14. Determine the smallest

value ofnso thatP(1≤Y)≥ 0 .70.

3.1.7. Let the independent random variablesX 1 andX 2 have binomial distribu-
tion with parametersn 1 =3,p=^23 andn 2 =4,p=^12 , respectively. Compute
P(X 1 =X 2 ).
Hint:List the four mutually exclusive ways thatX 1 =X 2 and compute the prob-
ability of each.
3.1.8.For this exercise, the reader must have access to a statistical package that
obtains the binomial distribution. Hints are given for R code, but other packages
canbeusedtoo.

(a)Obtain the plot of the pmf for theb(15, 0 .2) distribution. Using R, the follow-

ing commands return the plot:

x<-0:15; plot(dbinom(x,15,.2)~x)

(b)Repeat part (a) for the binomial distributions withn=15andwithp=

0. 10 , 0. 20 ,..., 0 .90. Comment on the shapes of the pmf’s aspincreases. Use

the following R segment: