Other Important Random Variables - Random Signals and Processes Primer with MATLAB - page 319

5.6.3 Binomial Distribution and Density

From ( 5.108 ) the binomial distribution is given as:

p k q nk uðx kÞ¼ X

F X ðxÞ¼ X

n n

n
k

P X ðk ; nÞuðx kÞ: (5.110)

k¼ 0 k¼ 0

The probabilities P X ( k ; n ) are called the probability mass function , as indicated
in Chap. 2 .
Keeping in mind that a binomial variable is a discrete variable, its corresponding
density function is:

p k q nk dðx kÞ¼ X

f X ðxÞ¼ X

n n

n
k

P X ðk; nÞdðx kÞ: (5.111)

k¼ 0 k¼ 0

The name “binomial” comes up from the probability ( 5.109 ) which itself
represents the n th degree of the Newton binomial:

¼ X

n

n

C n p k q nk ðp þ qÞ : (5.112)

k¼ 0

Taking into account ( 5.101 ), and using ( 5.109 ) from ( 5.112 ), we arrive at:

¼ 1 ¼ X ¼ X

n n

1 n

C n p k q nk P X ðk; nÞ:

(5.113)

k¼ 0 k¼ 0

The expression ( 5.113 ) proves that the binomial PDF satisfies the condition in
( 2.83 ).
Additionally, the expression ( 5.112 ) can be useful in calculating the characteris-
tic function, as shown below.

5.6.4 Characteristic Functions and Moments

The characteristic function of a binomial variable is:

f X ðoÞ¼Ef e joX g

¼ X e jok P X ðk ; nÞ¼ X ¼ X

n n n

k

e jok C n p k q nk C n ðp e jo q nk Þ : (5.114)

k¼ 0 k¼ 0 k¼ 0