Normal Random Variable - Random Signals and Processes Primer with MATLAB - page 227

1 1

ðx aÞ e ð x a Þ 2 a e ð x a Þ 2

1
2 p

1
2 p

¼ p d x þ p d x: (4.2)

2 b 2 2 b 2

b b

1 1

One can easily see that the first term in ( 4.2 ) can be expressed as:

1 1

ðx aÞ e ð x a Þ 2 ðx aÞ e ð x a Þ 2

1
2 p

1
2 p

p d x ¼ p d x

2 b 2 2 b 2

b b

1 0

1

ðx aÞ e ð x a Þ 2

1
2 p

p d x ¼ 0 : (4.3)

2 b 2

b

0

Similarly, using the PDF property ( 2.83 ) and ( 4.1 ), we have:

1 1

a e ð x a Þ 2

e ð x a Þ 2

1
2 p

1
2 p

p

d x ¼ a

p

d x ¼ a:

(4.4)

2 b 2 2 b 2

b b

1 1

From ( 4.1 )to( 4.4 ), we get:

EfXg¼m ¼ a:

(4.5)

Next we find the variance of the normal randomvariable X ,using( 2.344 ) and( 4.1 ):

1 1

2 e ð x m Þ 2

1
2 p

2

s 2

¼ ðx mÞ f X ðxÞ d x ¼ p ðx mÞ d x: (4.6)

2 b 2

b

1 1

By introducing the variable u

x m

u ¼

p b

(4.7)

in ( 4.6 ) we arrive at:

1 1

2 b 2
p

4 b 2
p

s 2 u 2 e u 2 d u ¼ u 2 e u 2 d u:

p p

¼ (4.8)

1 0