handout_week3_c

# handout_week3_c - Stochastic Signals and Systems Random...

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Stochastic Signals and Systems Random Variables Virginia Tech Fall 2008 Many-One Transformations If several values of x can be transformed into one value of y , we say that y = g ( x ) represents a many-one value. Thus with Y = X 2 , both x and - x go into the same value of y . It is merely necessary to add the probabilities of all the “infinitesimal events” ( x < X < x + dx ) that go into the same event ( y < Y < y + dy ) .

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Many-One Transformations The event ( y < Y < y + dy ) occurs when any one of the three events involving X occurs. These events will be mutually exclusive if dy is sufficiently small. Hence P [ y < Y < y + dy ] = P [ x 1 < X < x 1 + dx 1 ] + P [ x 2 < X < x 2 + dx 2 ] + P [ x 3 < X < x 3 + dx 3 ] and in terms of the pdf of X and Y , f Y ( y ) | dy | = f X ( x 1 ) | dx 1 | + f X ( x 2 ) | dx 2 | + f X ( x 3 ) | dx 3 | . Dividing by | dy | , we find the pdf of Y f Y ( y ) = f X ( x 1 ) dx 1 dy x 1 = g - 1 ( y ) + . . . + f X ( x k ) dx k dy x k = g - 1 ( y ) Example Let X be a Gaussian random variable with mean m = 0 and standard deviation σ = 1. Let Y = X 2 . Find the pdf of Y .
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