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handout_week3_b - Stochastic Signals and Systems Random...

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Stochastic Signals and Systems Random Variables Virginia Tech Fall 2008 Functions of a Random Variable We treat the problem of calculating the probability density function and the cumulative distribution function of a random variable Y that is a given function g ( X ) of a random variable X having a known probability density function f X ( x ) . 1 Y is also a random variable. 2 The probabilities with which Y takes on various values depend on the function g ( x ) and on f X ( x ) . 3 We assume the function g ( x ) has a unique value for all values of x .
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Discrete Random Variable The problem is simplest when the input random variable X is of discrete type, taking values in a countable set x 1 , x 2 , . . . with probabilities P [ X = x k ] , k = 1 , 2 , . . . which sum to 1. Then the random variable Y takes on only a countable set of values y 1 = g ( x 1 ) , y 2 = g ( x 2 ); . . . y k = g ( x k ); . . . If y = g ( x ) is a monotone function of x , it has a unique inverse, which we write as x = g - 1 ( y ) Then the probabilities of the various values of the random variable Y are simply P [ Y = y k ] = P [ X = g
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