Other Important Random Variables - Random Signals and Processes Primer with MATLAB - page 318

is a discrete random variable with only two discrete values 1 and 0, that has the
corresponding probabilities,

PfX ¼ 1 g¼p: (5.102)

PfX ¼ 0 g¼ 1 p ¼ q: (5.103)

5.6.2 What Is a Binomial Random Variable?

Consider n independent random Bernoulli exp e riments. As a result of each experi-
ment, either event A or its complement A occurs, with the corresponding
probabili tie s ( 5.100 ) and ( 5.101 ). The out co mes of repeated trials are sequences
of A and A where order of sequences A or A is arbitrary.

AAAA ... AAAA ... A A: (5.104)

The number of A events in n repeated Bernoulli trials can be any integer, from
0to n . A random variable associated with the number of the occurrences of event A
is said to be a binomial random variable , with the parameters, n and p , where p is a
probability of event A .
A binomial random variable X is a discrete random variable with the discrete
values x ¼ 0, ... , n . In order to describe this variable, we need the probability

PfX ¼ kg¼P X ðk ; nÞ; k ¼ 0 ; ... ; n: (5.105)

The total number of the combinations of k events A and ( n k ) events A in n
repeated experiments is equal to

¼

n
k

n ð n 1 Þ; ... ; ð n k þ 1 Þ
1 2 3 k

n !
k ! ðn kÞ !

C n ¼

¼ ; (5.106)

where C n is a binomial coefficient and the symbol “ N !” means factorial:
N ! ¼ 1 2 3 ðN 1 ÞN: (5.107)

Keeping in mind that the e xperiments are independent, each combination of k
events A and ( n k ) events A , will have a probability,

p k q nk ; (5.108)

resulting in:

p k q nk

n
k

P X ðk ; nÞ¼C n p k q nk ¼ : (5.109)

This expression is called a Bernoulli formula and Binomial probability law
[LEO94, p. 62].