Page 170 - Fundamentals of Probability and Statistics for Engineers

Functions of Random Variables 153

In order to conform with conditions stated in this section, we augment Equa-
tion (5.78) by some simple transformation such as

Z ˆ X 2 : 5:79†

The random variables Y and Z now play the role of Y 1 and Y 2 in Equation
(5.67) and we have

1

1
f YZ y; z†ˆ f X 1 X 2 ‰g y; z†; g y; z†ŠjJj; 5:80†
1
2
where
y

1
g y; z†ˆ ;
1 z

1
g y; z†ˆ z;
2

1
1 y
z ˆ :
2
J ˆ z z
0 1

Using specific forms of f (x 1 ) and f (x 2 ) given in Example (5.11), Equation
X 1 X 2
(5.80) becomes

y 1 2y 2
z

f YZ y; z†ˆ f f z† ˆ ;
z X 2 z 2z
X 1
z
y 2
z† 5:81†
ˆ ; for 0 y 2; and y z 2;
z 2
ˆ 0; elsewhere:
Finally, pdf f Y (y) is found by performing integration of Equation (5.81) with
respect to z:
2
Z 1 Z y 2
z†
f y†ˆ f YZ y; z† dz ˆ 2 dz;
Y

1 y z
ˆ 2 ‡ y ln y
1
ln 2†; for 0 y 2;
ˆ 0; elsewhere:
This result agrees with that given in Equation (5.47) in Example 5.11.

REFERENCE

Courant, R., 1937, Differential and Integral Calculus, Volume II, Wiley-Interscience,

New York.

TLFeBOOK