Applied Statistics and Probability for Engineers

5-3 TWO CONTINUOUS RANDOM VARIABLES 159

The probability that is determined as the integral over the

darkly shaded region in Fig. 5-9.

5-3.2 Marginal Probability Distributions

Similar to joint discrete random variables, we can find the marginal probability distributions

of Xand Yfrom the joint probability distribution.

0.003 1 316.73811.578 2 0.915

0.003 ca

1 e^3

0.003

be^4 a

1 e^1

0.001

bd

0.003 

1000

0

e0.003xe^4 e0.001x dx

 6 10 ^6 

1000

0

a

e0.002xe^4

0.002

b e0.001x dx

 6 10 ^6 

1000

0

°

2000

x

e0.002y^ dy¢ e0.001x^ dx

P 1 X 1000, Y 20002  

1000

0



2000

x

fXY 1 x, y 2 dy dx

X1000 and Y 2000

y

0 x

y

0 x

0

2000

1000

Figure 5-8 The joint probability

density function ofXandYis

nonzero over the shaded region.

Figure 5-9 Region of integration for

the probability thatX 1000 and Y

2000 is darkly shaded.

If the joint probability density function of continuous random variables Xand Yis

fXY(x, y), the marginal probability density functionsof Xand Yare

(5-16)

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) for which Yy

fX 1 x 2 

Rx

fXY 1 x, y 2 dy and fY 1 y 2 

Ry

fXY 1 x, y 2 dx

Definition

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