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: