Applied Statistics and Probability for Engineers

10-2

Proof Since U 1 and U 2 are independent chi-square random variables, their joint probability

distribution is

Using the method in Equation S5-4, define the new random variable MU 2. The inverse

solutions of

are

Therefore, the Jacobian is

Thus, the joint probability density function of Xand Mis

The probability density function of Fis

Substituting and , we obtain

f 1 x 2 

a

 1

 2 b

 1 2

x^1    2 ^1

21 ^1 ^22    2  a

 1

2

b  a

 2

2

b

(^) 
0
°
2 z
 1
 2 x^1
¢
1  1  22 2  1
ez 2 a
 1
 2 x^1 b
 1
dz
dm 2 a
 1
 2 x^1 b
 1
z dz
m
z^ a
 1
 2 x^1 b

a
 1
 2 xb
 1 2  1
a
 1
 2 b
21 ^1 ^22 2  a
 1
2
b  a
 2
2
b
(^) 
0
m^1 ^1 ^22 2 ^1 e^1 m^2231 ^1 ^22 x^14 dm

f 1 x 2 

0

f 1 x, m 2 dm

f 1 x, m 2 

a

 1

 2 mxb

 1 2  1

m^2 2 ^1 e^11 22 31 ^1 ^22 mxm^4 a

 1

 2 b^ m

2 ^1   2  a

 1

2

b 2 ^2 2 ^ a

 2

2

b

, 0

x, m

J

†

 1

 2 m

0

 1

 2 x

1

†



 1

 2 m

u 1 

 1

 2 mx^ and^ u^2 m

xa

u 1

 1 b^a

u 2

 2 b^ and^ mu^2

f 1 u 1 , u 22 

u 11   2 ^1 u 22 2 ^1

2 ^1   2  a

 1

2

b 2 ^2 2 ^ a

 2

2

b

e^1 u^1 u^22 2 , 0

u 1 , u 2

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