Applied Statistics and Probability for Engineers

5-4

EXAMPLE S5-3 Let Xbe a continuous random variable with probability distribution

Find the probability distribution of Y h(X) 2 X 4.

Note that y h(x) 2 x 4 is an increasing function of x. The inverse solution is x

u(y) (y4)2, and from this we find the Jacobian to be

Therefore, from S5-3 the probability distribution of Yis

We now consider the case where X 1 and X 2 are continuous random variables and we wish

to find the joint probability distribution of Y 1  h 1 (X 1 , X 2 ) and Y 2  h 2 (X 1 , X 2 ) where the trans-

formation is one to one. The application of this will typically be in finding the probability dis-

tribution of Y 1  h 1 (X 1 , X 2 ), analogous to the discrete case discussed above. We will need the

following result.

fY 1 y 2 

1 y 42

2

8

a

1

2

b

y 4

32

, 4 y 12

Ju¿ 1 y 2 dx dy 1
2.

fX 1 x 2 

x

8

, 0 x 4

Suppose that X 1 and X 2 are continuousrandom variables with joint probability distri-

bution and let Y 1  h 1 (X 1 ,X 2 ) and Y 2  h 2 (X 1 ,X 2 ) define a one-to-one

transformation between the points (x 1 , x 2 ) and (y 1 , y 2 ). Let the equations y 1 h 1 (x 1 ,

x 2 ) and y 2 h 2 (x 1 , x 2 ) be uniquely solved for x 1 and x 2 in terms of y 1 and y 2 as x 1 

u 1 (y 1 , y 2 ) and x 2 u 2 (y 1 , y 2 ). Then the joint probability of Y 1 and Y 2 is

(S5-4)

where Jis the Jacobianand is given by the following determinant:

and the absolute value of the determinant is used.

J`

x 1

 y 1 , x 1

 y 2

x 2

 y 1 , x 2

 y 2

`

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

fX 1 X 21 x 1 , x 22 ,

This result can be used to find the joint probability distribution of Y 1 and Y 2. Then

the probability distribution of Y 1 is

That is, (y 1 ) is the marginal probability distribution of Y 1.

EXAMPLE S5-4 Suppose that X 1 and X 2 are independent exponential random variables with

and fX 21 x 22  2 e^2 x^2 .Find the probability distribution of YX 1

 X 2.

fX 11 x 12  2 e^2 x^1

fY 1

fY 11 y 12  

   

fY 1 Y 2 1 y 1 , y 22 dy 2

fY 1 Y 21 y 1 , y 22 ,

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