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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