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Determining the PDF for a Function
of a Continuous Random Variable
In lecture, a formula was presented that can be used to determine the probability function
for y =
g
(x), where x is a continuous random variable, and
g
(
·
) is some function. The given
formula can also be found in Kay, equation (10.33). If, for a given value of y, there are
M
inverses of the function
g
(
·
), then the pdf for y is given by
p
y
(
y
) =
M
X
i
=1
p
x
‡
g

1
i
(
y
)
·
·
ﬂ
ﬂ
ﬂ
ﬂ
ﬂ
∂g

1
i
(
y
)
∂y
ﬂ
ﬂ
ﬂ
ﬂ
ﬂ
,
where
n
g

1
1
(
y
)
,...,g

1
M
(
y
)
o
are the
M
values of
x
such that
g
(
x
) =
y
.
There is another formula that can also be used to ﬁnd the pdf for y, and is in fact
equivalent to the formula given above:
p
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 Fall '09
 Jacobs

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