handout_week9_b - Stochastic Signals and Systems Multiple...

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Stochastic Signals and Systems Multiple Random Variables Virginia Tech Fall 2008 PDF of Sums of Independent RVs Let X 1 , X 2 , . . . , X n be n independent random variables. We’re interested in finding the pdf of S = X 1 + X 2 + . . . + X n If n = 2, the pdf of S = X 1 + X 2 is given by f S ( s ) = Z -∞ f X 1 , X 2 ( x , s - x ) dx = Z -∞ f X 1 ( x ) f X 2 ( s - x ) dx = f X 1 ( x ) * f X 2 ( x ) And the characteristic function of S is given by Φ S ( ω ) = E h e j ω S i = E h e j ω ( X 1 + X 2 ) i = E h e j ω X 1 e j ω X 2 i = E h e j ω X 1 i E h e j ω X 2 i = Φ X 1 ( ω X 2 ( ω )
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PDF of Sums of Independent RVs The previous equation states the well-known result that the Fourier transform of a convolution of two functions is equal to the product of the individual Fourier transforms. Now consider the sum of n independent random variables: S = X 1 + X 2 + . . . + X n . The characteristic function of S is Φ S ( ω ) = E h e j ω S i = E h e j ω ( X 1 + ... + X n ) i = E h e j ω X 1 i . . . E h e j ω X n i = Φ X 1 ( ω ) . . . Φ X n ( ω ) Thus the pdf of S can then be found by finding the inverse
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