Applied Statistics and Probability for Engineers

5-6

and. From Equation S5-4, the joint probability density function of Y 1 and Y 2 is

Therefore, the marginal probability density function of Y 1 is

The notation is simpler if the variable of integration y 2 is replaced with xand y 1 is replaced

with y. Then the following result is obtained.

fY 11 y 12  

   

fX 1 1 y 1 y 22 fX 2 1 y 22 dy 2

fY 1 Y 21 y 1 , y 22 f (^) X 11 y 1 y 22 f (^) X 21 y 22
ƒJƒ 1
If X 1 and X 2 are independent random variables with probability density functions
(x 1 ) and (x 2 ), respectively, the probability density function of YX 1 X 2 is

fY 1 y 2  (S5-5)

   

fX 1 1 yx 2 f (^) X 2 1 x 2 dx
fX 1 fX 2
Convolution
of X 1 and X 2
The probability density function of Yin Equation S5-5 is referred to as the convolutionof the
probability density functions for X 1 and X 2. This concept is commonly used for transforma-
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 1 and X 2. A similar result can be obtained for discrete random variables with the
integral replaced with a sum.
In some problems involving transformations, we need to find the probability distribution
of the random variable Yh(X) when Xis a continuous random variable, but the transforma-
tion is not one to one. The following result is helpful.
Suppose that Xis a continuousrandom variable with probability distribution fX(x),
and Yh(X) is a transformation that is not one to one. If the interval over which X
is defined can be partitioned into mmutually exclusive disjoint sets such that each of
the inverse functions x 1 u 1 (y), x 2 u 2 (y), , xmum(y) of yu(x) is one to
one, the probability distribution of Yis
(S5-6)
where Jiu¿i 1 y 2 , i 1, 2,p, mand the absolute values are used.
fY 1 y 2  a
m
i 1
fX 3 ui 1 y 24 0 Ji 0
p
To illustrate how this equation is used, suppose that Xis a normal random variable with
mean and variance^2 , and we wish to show that the distribution of Y 1 X 22
^2 is a
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