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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.
You will receive feedback from your instructor and TA directly on this page. Other students are welcome to comment/discuss/point out mistakes/ask questions too!
Answer 1
I'm not sure what I'm missing on part d) - I know $ Y_1 $ and $ Y_2 $ have to have a covariance of zero and that the correlation coefficient is zero for independence. So I end up with $ Y=M(X-\mu) $ and $ E(Y_1Y_2)=E(Y_1)E(Y_2) $ but I don't know where to go from there.
- Try to diagonalize $ \Sigma $. Did you take linear algebra? -pm
Answer 2
Write it here.
Answer 3
Write it here.
