Fundamentals of Probability and Statistics for Engineers

Note that, for a more complicated jpdf, one needs to carry out the volume
integral for volume calculations.

As an exercise, let us determine the joint probability distribution function
and the marginal density functions of random variables X and Y defined in
Example 3.7.
The JPDF of X and Y is obtained from Equation (3.25). It is clear that

Within the region (0, 0) (x, y) (60, 60), we have

For marginal density functions, Equations (3.28) and (3.29) give us

y

x

60

60

010

R

10

y–x=10

x–y=10

Figure 3. 16 Region R in Example 3.7

Random Variables and Probability D istributions 59

RR

RfXY^9 x,y)dxdy

FXY…x;y†ˆ

0 ; for…x;y†<… 0 ; 0 †;

1 ; for…x;y†>… 60 ; 60 †:





FXY…x;y†ˆ

Z y

0

Zx

0

1

3600



dxdyˆ

xy

3600

:

fX…x†ˆ

Z 60

0

1

3600



dyˆ

1

60

; for 0x 60 ;

0 ; elsewhere:

8

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