Other Important Random Variables - Random Signals and Processes Primer with MATLAB - page 324

As a consequence most of the values of the random variable are concentrated
around the mean value np , which is a well-known property of normal random
variables. Therefore, in this case, the probabilities ( 5.109 ) can be approximated
with a normal variable with the parameters:

m ¼ np
s 2

(5.131)

¼ npq:

It follows:

2

e ð k np Þ f X ðxÞ ¼ X

k¼ 0

C n p k q nk dðxkÞ X

n n

1
2 pnpq

p

2 npq

dðxkÞ: (5.132)

k¼ 0

Note that the normal density in ( 5.132 ) approximates the probability mass
function ( 5.109 )
In this way, we can use the functions shown in Sect. 4.2.2 to find the approximate
probabilities for a binomial random variable:

Pfkk 1 ; ng ¼ X

k 1

1
2

1 þ erf k 1 m C n p k q nk

s 2

p

: (5.133)

k¼ 0

Similarly,

Pfk 2 kk 3 ; ng ¼ X

k 3

1
2

erf k 3 m

s 2 p erf k 2 m

C n p k q nk

s 2

p

: (5.134)

k¼k 2

Example 5.6.3 In n ¼ 60 independent trials, a binary event A occurs with a
probability of p ¼ 0.3. Find the probability that event A occurs greater than or
equal to 20 and less than or equal to 40 times.

Solution The number of occurrences of event A is a binomial variable with a mean
value and variance of:

m ¼ np ¼ 60 0 : 3 ¼ 18 : (5.135)

s 2 ¼ npq ¼ 60 0 : 3 ð 1 0 : 3 Þ ¼ 12 : 6 : (5.136)

The corresponding normal density approximation is given as:

f X ðxÞ X
k¼ 0

n

e ðk 18 Þ 2

1
2 p 12

p

6 dðxkÞ:

(5.137)

2 12 :

: 6

From ( 5.134 ), we get:

Pf 20 k 40 ; 60 g

"

#

1
2

40 18
2 12

20 18
2 12

p p

erf erf ¼ 0 : 7134 : (5.138)

: 6 : 6