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:
- The number of misprints on a page (or a group of pages) of a book.
- The number of people in a community living to 100 years of age.
- The number of wrong telephone numbers that are dialed in a day.
- The number of transistors that fail on their first day of use.
- The number of customers entering a post office on a given day.