Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

3.1. The Binomial and Related Distributions 157

which is not a simple calculation. Most statistical packages provide procedures to

calculate binomial probabilities. In R, ifYisb(n, p) then the cdf ofYis computed

asP(Y≤y)=pbinom(y,n,p). Hence, for our example, using R we compute the

P(Y≥16) as

P(Y≥16) = 1−P(Y≤15) = 1−pbinom(15, 60 , 1 /6) = 0. 0338.

The R functiondbinomcomputes the pmf of a binomial distribution. For instance,

to compute the probability thatY= 11, we use the R code:dbinom(11,60,1/6),

which computes to 0.1246.

The mgf of a binomial distribution is easily obtained as follows:

M(t)=

∑

x

etxp(x)=

∑n

x=0

etx

(

n

x

)

px(1−p)n−x

=

∑n

x=0

(

n

x

)

(pet)x(1−p)n−x

=[(1−p)+pet]n

for all real values oft.Themeanμand the varianceσ^2 ofXmay be computed
fromM(t). Since

M′(t)=n[(1−p)+pet]n−^1 (pet)

and

M′′(t)=n[(1−p)+pet]n−^1 (pet)+n(n−1)[(1−p)+pet]n−^2 (pet)^2 ,

if follows that

μ=M′(0) =np

and

σ^2 =M′′(0)−μ^2 =np+n(n−1)p^2 −(np)^2 =np(1−p).

SupposeYhas theb(60, 1 /6) distribution as discussed in Example 3.1.1. Then

E(Y) = 60(1/6) = 10 and Var(Y) = 60(1/6)(5/6) = 8. 33

Example 3.1.2.If the mgf of a random variableXis

M(t)=(^23 +^13 et)^5 ,

thenXhas a binomial distribution withn=5andp=^13 ;thatis,thepmfofXis

p(x)=

{ ( 5

x

)( 1

3

)x( 2

3

) 5 −x

x=0, 1 , 2 ,..., 5

0elsewhere.

Hereμ=np=^53 andσ^2 =np(1−p)=^109.