Introduction to Probability and Statistics for Engineers and Scientists

5.1The Bernoulli and Binomial Random Variables 147

Var(X)=

∑n

i= 1

Var(Xi) since theXiare independent

=np(1−p)

IfX 1 andX 2 are independent binomial random variables having respective parameters
(ni,p),i=1, 2, then their sum is binomial with parameters (n 1 +n 2 ,p). This can most
easily be seen by noting that becauseXi,i=1, 2, represents the number of successes inni
independent trials each of which is a success with probabilityp, thenX 1 +X 2 represents
the number of successes inn 1 +n 2 independent trials each of which is a success with
probabilityp. Therefore,X 1 +X 2 is binomial with parameters (n 1 +n 2 ,p).

5.1.1 Computing the Binomial Distribution Function

Suppose thatXis binomial with parameters (n, p). The key to computing its distribution
function

P{X≤i}=

∑i

k= 0

(

n

k

)

pk(1−p)n−k, i=0, 1,...,n

is to utilize the following relationship betweenP{X=k+ 1 }andP{X=k}:

P{X=k+ 1 }=

p

1 −p

n−k

k+ 1

P{X=k} (5.1.4)

The proof of this equation is left as an exercise.

EXAMPLE 5.1e LetXbe a binomial random variable with parametersn=6,p=.4. Then,
starting withP{X= 0 }=(.6)^6 and recursively employing Equation 5.1.4, we obtain

P{X= 0 }=(.6)^6 =.0467

P{X= 1 }=^4661 P{X= 0 }=.1866

P{X= 2 }=^4652 P{X= 1 }=.3110

P{X= 3 }=^4643 P{X= 2 }=.2765

P{X= 4 }=^4634 P{X= 3 }=.1382

P{X= 5 }=^4625 P{X= 4 }=.0369

P{X= 6 }=^4616 P{X= 5 }=.0041. ■

The text disk uses Equation 5.1.4 to compute binomial probabilities. In using it, one enters
the binomial parametersnandpand a valueiand the program computes the probabilities
that a binomial (n, p) random variable is equal to and is less than or equal toi.