handout_week8_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 Linear Transformation of Gaussian RVs Let X = ( X 1 ,..., X n ) T be jointly Gaussian and define Y = ( Y 1 ,..., Y n ) T by Y = A X where A is an n × n invertible matrix. By making a transformation of variables we have: f Y ( y ) = f X ( A - 1 y ) | A | = exp n - 1 2 ( A - 1 y - m ) T K - 1 ( A - 1 y - m ) o ( 2 π ) n / 2 | A || K | 1 / 2 = exp n - 1 2 ( y - n ) T C - 1 ( y - n ) o ( 2 π ) n / 2 | C | 1 / 2 Thus, Y is jointly Gaussian, with mean vector n = A m and covariance matrix C = AKA T .
Background image of page 1

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

View Full DocumentRight Arrow Icon
The random variables X and Y are jointly Gaussian with E [ X ] = 10 E [ Y ] = 0 σ 2 X = 4 σ 2 Y = 1 ρ X , Y = 0 . 5 Find the joint density function of the random variables Z = X + Y W = X - Y Test 2, Fall’06 The random variables X 1 , X 2 , X 3 are jointly Gaussian. All have zero-mean and unit variance. Their correlations are given by E [ X 1 X 2 ] = 1 3 E [ X 1 X 3 ] = - 1 E [ X 2 X 3 ] = 1 3 Obtain an explicit expression for the joint probability
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_week8_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