# three i.i.d. Gaussian random variables - joint pdf

• Nov 12th 2011, 06:18 AM
three i.i.d. Gaussian random variables - joint pdf
1) you have a 3x1 vector x of three i.i.d. Gaussian random variables.
you have a matrix A which is k x 3, where k=2, 3, 4, and in all cases the
coefficients of A are such that its rank is the minimum between k and 3.
Write the joint p.d.f. of y=Ax for al the values of k.
(Note: k=2,3 are easy but k=4 requires some thought)

2) Let x be a vector of variables, A a matrix and y a given vector. What is the solution of min_x ||Ax-y||^2
• Nov 12th 2011, 10:32 PM
CaptainBlack
Re: three i.i.d. Gaussian random variables - joint pdf
Quote:

Originally Posted by mabdulahad
1) you have a 3x1 vector x of three i.i.d. Gaussian random variables.
you have a matrix A which is k x 3, where k=2, 3, 4, and in all cases the
coefficients of A are such that its rank is the minimum between k and 3.
Write the joint p.d.f. of y=Ax for al the values of k.
(Note: k=2,3 are easy but k=4 requires some thought)

The covariance matrix of $\displaystyle X=[x_1,x_2,x_3]'$ is the diagonal matrix $\displaystyle \Sigma$ with $\displaystyle \sigma_1^2, \sigma_2^2, \sigma_3^3$ down the diagonal.

$\displaystyle \text{CoVar}(y)=\text{E}((y-\overline{y})(y-\overline{y})')=\text{E}(A(x-\overline{x})(x-\overline{x})'A')=A \Sigma A'$

CB
• Nov 13th 2011, 03:29 AM
Re: three i.i.d. Gaussian random variables - joint pdf
but what is the joint probability for k=2 , 3 and 4??? can you give detail solution?
• Nov 13th 2011, 06:34 AM
CaptainBlack
Re: three i.i.d. Gaussian random variables - joint pdf
Quote:

Originally Posted by mabdulahad
but what is the joint probability for k=2 , 3 and 4??? can you give detail solution?

The joint distribution is the multivariate normal with mean:

$\displaystyle A\overline{x}$

and covariance matrix:

$\displaystyle R=A \Sigma A'$

In the case k=4 the multivariate normal distribution is degenerate, see: Multivariate normal distribution - Wikipedia, the free encyclopedia

CB