Other Important Random Variables - Random Signals and Processes Primer with MATLAB - page 325

5.7 Poisson Random Variable

5.7.1 Approximation of Binomial Variable

Consider the asymptotic behavior of a binomial random variable if n is very high in
value and the probability of p is very small such that the mean value of a binomial
variable,

X ¼ k ¼ np (5.139)

has a moderate value.
From ( 5.139 ), we write:

k
n :

p ¼ (5.140)

The obtained value ( 5.140 ) is placed into the expression P X ( k ; n )in( 5.109 ), and n
is assumed to go to 1 , resulting in:

n k

nk nk

n
k

k k k

n

n !
k ! ðnkÞ !

k
n

lim
n!1

P X ðk; nÞ¼ lim
n!1

1 ¼ lim

n!1

n k 1

k n

k k
k !

n ð n 1 Þ:::ð n k þ 1 Þ
n k

k
n

k
n

¼ lim

n!1

1 1

:

k n

k k
k !

n
n

n 1 nkþ 1

n

k
n

k
n

¼ lim

n!1

n

1 1

n

k k
k !

k
n

¼ lim

n!1

1

(5.141)

Knowing that

n

k
n

¼ e k

lim
n!1

1 ; (5.142)

it follows that

k k
k !

e k lim

n!1

P X ðk; nÞ¼PfX ¼ kg¼ : (5.143)

The obtained expression for probability is called the Poisson formula and it is a
probability mass function (see Sect. 2.2.1 ) ofa Poisson random variable , where k is
the parameter.