Fundamentals of Probability and Statistics for Engineers

Answer: according to Equation (4.46),

U sing Equation (4.52), we have

and

The results for the mean and variance are the same as those obtained in
Examples 4.1 and 4.5.

Ex ample 4. 15. Problem: repeat the above when X is exponentially distributed
with density function

Answer: the characteristic function X (t) in this case is

The moments are

which agree with the moment calculations carried out in Examples 4.2 and 4.6.

100 Fundamentals of Probability and Statistics for Engineers

X…t†ˆ

Xn

kˆ 0

ejtk

n

k



pk… 1 p†nk

ˆ

Xn

kˆ 0

n

k



…pejt†k… 1 p†nk

ˆ‰pejt‡… 1 p†Šn:

… 4 : 53 †

1 ˆ

1

j

d

dt

‰pejt‡… 1 p†Šn

tˆ 0

ˆn‰pejt‡… 1 p†Šn^1 …pejt†

tˆ 0

ˆnp;

2 ˆ

d^2

dt^2

‰pejt‡… 1 p†Šn

tˆ 0

ˆnp‰…n 1 †p‡ 1 Š;

^2 Xˆ 2  21 ˆnp‰…n 1 †p‡ 1 Šn^2 p^2 ˆnp… 1 p†:

fX…x†ˆ

aeax; forx 0 ;

0 ; elsewhere:





X…t†ˆ

Z 1

0

ejtx…aeax†dxˆa

Z 1

0

e…ajt†xdxˆ

a

ajt

: … 4 : 54 †

1 ˆ

1

j

d

dt

a

ajt

^

tˆ 0

ˆ

1

j

ja

…ajt†^2

"

tˆ 0

ˆ

1

a

;

2 ˆ

d^2

dt^2

a

ajt

 (^)
tˆ 0

ˆ

2

a^2

;

^2 Xˆ 2  21 ˆ

1

a^2

;