Revision as of 09:57, 20 March 2013

Practice Problem: Various Questions about a 2D Gaussian

Let

$ X=\left( \begin{array}{l} X_1\\ X_2 \end{array} \right) $

be a two-dimensional Gaussian random variable with mean $ \mu $ and standard deviation matrix $ \Sigma $ given by

$ \mu=\left( \begin{array}{c} -1\\ 2 \end{array} \right) , \Sigma=\left( \begin{array}{cc} 3 & 1 \\ 1 & 3 \end{array} \right) $

a) Write the pdf of X using matrix notation.

b) Write the pdf of X without matrix or vector.

c) Find the marginal pdf for $ X_1 $.

d) Find a matrix M such that the vector $ Y=M(X-\mu) $ consists of independent random variables.

e) Find the joint pdf of Y.

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