Normal Random Variable - Random Signals and Processes Primer with MATLAB - page 236

Similarly, from ( 4.23 ) and ( 4.25 ), it follows:

ð

:

u

2
p

x m
p s

e u 2 d u ¼ erf ðuÞ¼ erf

p

(4.28)

0

Finally, from ( 4.26 )to( 4.28 ), we have:

1
2

x m
p s F X ðxÞ¼ 1 þ erf : (4.29)

Using ( 4.29 ), the probability P { X x 1 } can be written as:

1
2

x 1 m
p s

PfX x 1 g¼F X ðx 1 Þ¼

1 þ erf

:

(4.30)

Similarly, from ( 2.46 ) and ( 4.29 ), we have:

Pfx 1 X x 2 g¼F X ðx 2 ÞF X ðx 1 Þ

1
2

x 2 m 1

2

x 1 m
p s ¼ 1 þ erf

p s

1 þ erf

erf

1
2

x 2 m
p s

x 1 m
p s ¼ erf : (4.31)

We often need to find the probability that the normal variable is around its
mean value in the interval [ m ks , m+ks ], where k is any real number. Using
( 4.24d ) and ( 4.31 ), where,

x 2 ¼ m þ ks;
x 1 ¼ m ks;

we find the desired probability as:

1
2

m þ ks m
p s

m ks m
p s

Pfm ks X m þ ksg¼ erf erf

:

k
2

¼ erf p (4.33)

Example 4.2.1 Given a normal random variable with a mean value of 5 and the
mean squared value of 89, find the probability that the random variable is less than
10 and that is in the interval [ 2, 8] calculating the erf function using the MATLAB
file erf( x ) .