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1 Finding the distribution (pdf) of functions of a random variables: Given: the new random variable Y is a function of another random variable X with a known pdf ( ) X p x . That is, () Yg X = . Goal: find the pdf of the random variable Y using the given knowledge on the pdf of X and the mapping that connects the random variable Y with the X . That is ( ) ( ) ? find the expression and its connection to the g mapping and the pdf ( ) Y X p yp x =⋅ . The simplest case : If the mapping is one-to-one (illustrated by the following figure) Results: 1 ( ) with ' ' X Y xg y p x dg x py g gd x = == Rationale: { } { } . P P eqv YX Yy p y y Xx p x x δδ ∆∆ ∈≈ →∈

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2 Extension to more general cases : If the mapping is many-to-one (as illustrated by the following figure) Results: 1 2 k=1 () ( ) with ' , ' kk Xk Y xg y px dg x py g gd x = == 12 Note here the mapping is 2-to-1: ( ) ( ) yg x g x = = Rationale: { } { } . 1 2 P P ,O R , eqv Y XX Yy p y y Xx p x x p x x δδ δ ∆∆ ∈≈ →∈ + Result
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