Fundamentals of Probability and Statistics for Engineers

which is expected. It gives the relationship between the joint jpmf and the
conditional mass function. As we will see in Example 3.9, it is sometimes more
convenient to derive joint mass functions by using Equation (3.35), as condi-
tional mass functions are more readily available.
If random variables X and Y are independent, then the definition of inde-
pendence, Equation (2.16), implies

and Equation (3.35) becomes

Thus, when, and only when, random variables X and Y are independent, their
jpmf is the product of the marginal mass functions.
Let X be a continuous random variable. A consistent definition of the
conditional density function of X given Y is the derivative of
its corresponding conditional distribution function. Hence,

where FX Y (xy) is defined in Equation (3.33). To see what this definition leads
to, let us consider

In terms of jpdf fX Y (x, y), it is given by

By setting nd and by taking the limit

Equation (3.40) reduces to

provided that fY (y) 0.

62 Fundamentals of Probability and Statistics for Engineers

pXY…xjy†ˆpX…x†; … 3 : 36 †

pXY…x;y†ˆpX…x†pY…y†: … 3 : 37 †

ˆy,fXY 9 xjy),

fXY…xjy†ˆ

dFXY…xjy†

dx

; … 3 : 38 †

j

P…x 1 <Xx 2 jy 1 <Yy 2 †ˆ

P…x 1 <Xx 2 \y 1 <Yy 2 †

P…y 1 <Yy 2 †

: … 3 : 39 †

P…x 1 <Xx 2 jy 1 <Yy 2 †ˆ

Zy 2

y 1

Zx 2

x 1

fXY…x;y†dxdy

Z y 2

y 1

Z 1

1

fXY…x;y†dxdy

ˆ

Zy 2

y 1

Zx 2

x 1

fXY…x;y†dxdy

Z y 2

y 1

fY…y†dy: … 3 : 40 †

x 1 ˆ1,x 2 ˆx,y 1 ˆy,a y 2 ˆy‡y,
y!0,

FXY…xjy†ˆ

Zx

1

fXY…u;y†du

fY…y†

; … 3 : 41 †

6ˆ