Applied Statistics and Probability for Engineers

160 CHAPTER 5 JOINT PROBABILITY DISTRIBUTIONS

(5-17)

and

where RXdenotes the set of all points in the range of (X, Y) for which Xxand

RYdenotes the set of all points in the range of (X, Y)



R

1 xX 22 fXY 1 x, y 2 dx dy

V 1 X 2  (^2) x (^) 


1 xX 22 fX 1 x 2 dx 



1 xX 22 £

Rx

fXY 1 x, y 2 dy§ dx



R

xfXY 1 x, y 2 dx dy

E 1 X 2 X



xfX 1 x 2 dx 



x £

Rx

fXY 1 x, y 2 dy§ dx

Mean and

Variance from

Joint

Distribution

A probability involving only one random variable, say, for example,

can be found from the marginal probability distribution of Xor from the joint probability

distribution of Xand Y. For example, P(a X b) equals P(a X b,  Y ).

Therefore,

Similarly, E(X) and V(X) can be obtained directly from the joint probability distribution ofX

and Yor by first calculating the marginal probability distribution of X. The details, shown in

the following equations, are similar to those used for discrete random variables.

P 1 aXb 2 

b

a



Rx

fXY 1 x, y 2 dy dx

b

a

°

Rx

fXY^1 x, y^2 dy¢dx

b

a

fX 1 x 2 dx

P 1 aXb 2 ,

EXAMPLE 5-16 For the random variables that denote times in Example 5-15, calculate the probability that Y

exceeds 2000 milliseconds.

This probability is determined as the integral of fXY(x, y) over the darkly shaded region

in Fig. 5-10. The region is partitioned into two parts and different limits of integration are de-

termined for each part.

 

2000

°

x

610 ^6 e0.001x0.002y^ dy¢ dx

P 1 Y

20002  

2000

0

°

2000

610 ^6 e0.001x0.002y dy¢ dx

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