Page 218 - Applied Statistics And Probability For Engineers

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5-6

and ƒ J ƒ 1 . From Equation S5-4, the joint probability density function of Y and Y is
1
2
1 y 2 2
f Y 1 Y 2 1 y 1 , y 2 2 f X 1 1y 1 y 2 2 f X 2
Therefore, the marginal probability density function of Y 1 is

1y 2 f 1 y y 2 f 1 y 2 dy
f Y 1 1 X 1 1 2 X 2 2 2

is replaced with x and y is replaced
The notation is simpler if the variable of integration y 2 1
with y. Then the following result is obtained.

Convolution
of X 1 and X 2 If X 1 and X 2 are independent random variables with probability density functions
f (x 2 ), respectively, the probability density function of Y X 1 X 2 is
X 1 (x 1 ) and f X 2

1 y2 f 1x2 dx (S5-5)
f Y X 1 1 y x2 f X 2

The probability density function of Y in Equation S5-5 is referred to as the convolution of the
probability density functions for X and X . This concept is commonly used for transforma-
2
1
tions (such as Fourier transformations) in mathematics. This integral may be evaluated nu-
merically to obtain the probability density function of Y, even for complex probability density
functions for X and X . A similar result can be obtained for discrete random variables with the
2
1
integral replaced with a sum.
In some problems involving transformations, we need to find the probability distribution
of the random variable Y h(X) when X is a continuous random variable, but the transforma-
tion is not one to one. The following result is helpful.

Suppose that X is a continuous random variable with probability distribution f X (x),
and Y h(X) is a transformation that is not one to one. If the interval over which X
is defined can be partitioned into m mutually exclusive disjoint sets such that each of
u (y), x u (y), p , x u (y) of y u(x) is one to
the inverse functions x 1 1 2 2 m m
one, the probability distribution of Y is
m
f 1 y2 a f 3u 1 y24 0 J i 0 (S5-6)
X

i
Y
i 1
where J u¿ 1 y2 , i 1, 2, p , m and the absolute values are used.
i
i
To illustrate how this equation is used, suppose that X is a normal random variable with
2
2
mean and variance , and we wish to show that the distribution of Y 1X 2
2 is a