Other Important Random Variables - Random Signals and Processes Primer with MATLAB - page 323

The corresponding probabilities are:

P X ð 0 ; 4 Þ¼ 0 : 6567

P X ð 1 ; 4 Þ¼ 0 : 2916

P X ð 2 ; 4 Þ¼ 0 : 0486 : (5.128)

P X ð 3

4 Þ¼ 0

:

; 0036

P X ð 4

4 Þ¼ 0

:

; 001

The density function is given in Fig. 5.16 .

5.6.5 Approximation of Binomial Variable

For high values of n , the calculation of the probabilities P X ( k ; n ) is time consuming.
Instead, it is helpful to use the asymptotic approximations for the binomial random
variable.
We will consider two cases:

Case 1 The probability p is finite and n is very high.

Case 2 The probability p is very small, while n is very high, resulting in a finite
value of np .

In this case, the approximation is the Poisson random variable, which will be
considered in Sect. 5.7 .
Let us now consider Case 1 .
It follows that

np 1 (5.129)

and

npq 1 : (5.130)

0.7

0.6

0.5
0.4

0.3

0.2
0.1

0 0

0.5 1 1.5 2

X

2.5 3 3.5 4

Fig. 5.16 Binomial density function