handout_week9_b - Stochastic Signals and Systems Multiple...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
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 )
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

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.

Page1 / 5

handout_week9_b - Stochastic Signals and Systems Multiple...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online