Introduction to Probability and Statistics for Engineers and Scientists

150 Chapter 5: Special Random Variables

Evaluating att=0 gives that

E[X]=φ′(0)=λ

Var(X)=φ′′(0)−(E[X])^2

=λ^2 +λ−λ^2 =λ

Thus both the mean and the variance of a Poisson random variable are equal to the
parameterλ.
The Poisson random variable has a wide range of applications in a variety of areas because
it may be used as an approximation for a binomial random variable with parameters (n,p)
whennis large andpis small. To see this, suppose thatXis a binomial random variable
with parameters (n,p) and letλ=np. Then

P{X=i}=

n!

(n−1)!i!

pi(1−p)n−i

=

n!

(n−1)!i!

(

λ

n

)i(

1 −

λ

n

)n−i

=

n(n−1)...(n−i+1)

ni

λi

i!

(1−λ/n)n

(1−λ/n)i

Now, fornlarge andpsmall,

(

1 −

λ

n

)n

≈e−λ

n(n−1)...(n−i+1)

ni

≈ 1

(

1 −

λ

n

)i

≈ 1

Hence, fornlarge andpsmall,

P{X=i}≈e−λ

λi

i!

In other words, ifnindependent trials, each of which results in a “success” with probability
p, are performed, then whennis large andpsmall, the number of successes occurring is
approximately a Poisson random variable with meanλ=np.
Some examples of random variables that usually obey, to a good approximation, the
Poisson probability law (that is, they usually obey Equation 5.2.1 for some value ofλ) are:

1. The number of misprints on a page (or a group of pages) of a book.


2. The number of people in a community living to 100 years of age.


3. The number of wrong telephone numbers that are dialed in a day.


4. The number of transistors that fail on their first day of use.


5. The number of customers entering a post office on a given day.