handout_week7_a

# handout_week7_a - Stochastic Signals and Systems Multiple...

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Stochastic Signals and Systems Multiple Random Variables Virginia Tech Fall 2008 Two Functions of Two RVs Suppose X and Y are two random variables with joint pdf f X , Y ( x , y ) . Given two functions g 1 ( x , y ) and g 2 ( x , y ) , deﬁne two new random variables V = g 1 ( X , Y ) W = g 2 ( X , Y ) How does one determine their joint pdf f V , W ( v , w ) ?

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Two Functions of Two RVs One possible procedure for determining f V , W ( v , w ) is to calculate the joint cdf of V = g 1 ( X , Y ) and W = g 2 ( X , Y ) by determining the probability F V , W ( v , w ) = P [ V v , W w ] = P [ g 1 ( X , Y ) v , g 2 ( X , Y ) w ] = Z Z R ( v , w ) f X , Y ( x , y ) dxdy where R ( v , w ) is the region in the xy plane such that the inequalities g 1 ( x , y ) v and g 2 ( x , y ) w are simultaneously satisﬁed. Example Suppose X and Y are independent random variables. Deﬁne V = min ( X , Y ) and W = max ( X , Y ) . Find f V , W ( v , w ) .
Two Functions of Two RVs Consider the elementary event

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## This note was uploaded on 05/05/2010 for the course ECECS 5605 taught by Professor Dasilver during the Fall '08 term at Virginia Tech.

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handout_week7_a - Stochastic Signals and Systems Multiple...

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