{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

handout_week8_b

# handout_week8_b - Stochastic Signals and Systems Multiple...

This preview shows pages 1–4. Sign up to view the full content.

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 .

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

View Full Document
Example 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 density function of Y 1 = X 1 + 3 X 2 + 2 X 3 Y 2 = - 4 X 1 + X 2 - 2 X 3
Joint Characteristic Function

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 5

handout_week8_b - Stochastic Signals and Systems Multiple...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online