Fundamentals of Probability and Statistics for Engineers

However, notice that, since X is continuous, P(X x) 0 if x is a point of
continuity in the distribution of X. Hence, using Equation (4.47),

The above defines the probability distribution function of X. Its derivative
gives the inversion formula

and we have Equation (4.58), as desired.

The inversion formula when X is a discrete random variable is

A proof of this relation can be constructed along the same lines as that given
above for the continuous case.

Proof of Equation (4.64):first note the standard integration formula:

Replacing a by X x and taking the limit as we have a new random
variable Y, defined by

The mean of Y is given by

Expectations and Moments 103

EfYgˆP…X<x†ˆFX…x†

ˆ

1

2



j

2 

Z 1

1

1 Efej…Xx†tg

t

dt … 4 : 62 †

ˆ

1

2



j

2 

Z 1

1

1 ejtxX…t†

t

dt:

ˆ ˆ

pX…x†ˆlim

u!1

1

2 u

Zu

u

ejtxX…t†dt: … 4 : 64 †

1

2 u

Zu

u

ejatdtˆ

sinau

au

; fora6ˆ 0 ;

1 ; foraˆ 0 :

8

<

:

… 4 : 65 †

 u!1,

fX…x†ˆ

1

2 

Z 1

1

ejtxX…t†dt; … 4 : 63 †

Yˆlim

u!1

1

2 u

Z u

u

ej…Xx†tdtˆ

0 ; forX6ˆx;

1 ; forXˆx:

EfYgˆ… 1 †P…Xˆx†‡… 0 †P…X6ˆx†ˆP…Xˆx†; … 4 : 66 †