Applied Statistics and Probability for Engineers

5-5

The joint probability distribution of X 1 and X 2 is

because X 1 and X 2 are independent. Let Y 1 h 1 (X 1 , X 2 ) and Y 2 h 2 (X 1 , X 2 ) 

X 1 X 2. The inverse solutions of y 1  and y 2 x 1 x 2 are and

, and it follows that

Therefore

and from Equation S5-4 the joint probability distribution of Y 1 and Y 2 is

for y 1

 0, y 2

 0. We need to find the distribution of. This is the marginal prob-

ability distribution of Y 1 , or

An important application of Equation S5-4 is to obtain the distribution of the sum of two

independentrandom variables X 1 and X 2. Let Y 1 X 1 X 2 and let Y 2 X 2. The inverse so-

lutions are x 1 y 1 y 2 and x 2 y 2. Therefore,

x 2

y 1

 0

x 2

y 2

 1

x 1

y 1

 1

x 1

y 2

 1



1

11 y 122

, y 1

0



0

4 e^2 y^2 3 y 2
11 y 1224 dy 2

fY 11 y 12 

0

fY 1 Y 21 y 1 , y 22 dy 2

Y 1 X 1
X 2

 4 e^2 y^2 y 2
11 y 122

 4 e^23 y^1 y^2

11 y^12 y^2

11 y^124 `

y 2

11 y 122

`

fY 1 Y 2 1 y 1 , y 22 fX 1 X 2 3 u 11 y 1 , y 22 , u 21 y 1 , y 2240 J 0

J∞

y 2

11 y 122

y 1

11 y 12

y 2

11 y 122

1

11 y 12

∞

y 2

11 y 122

x 2

y 1

y 2 c

 1

11 y 122

d,

x 2

y 2

c

1

11 y 12

d

x 1

y 1

y 2 c

1

11 y 122

d,

x 1

y 2

c

y 1

11 y 12

d

x 2 y 2
11 y 12

x 1
x 2 x 1 y 1 y 2
11 y 12

X 1
X 2

fX 1 X 2 1 x 1 , x 22  4 e^21 x^1 x^22 , x 1 0, x 2  0

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