Question

Random variables \(X\) and \(Y\) have joint probability mass function (PMF):

\(P_{X, Y}\left(x_{k}, y_{j}\right)=P\left(X=x_{k}, Y=y_{j}\right)= \begin{cases}\frac{1}{20}\left|x_{k}+y_{j}\right|, & x_{k}=-1,0,1 ; y_{j}=-3,0,3 \\ 0, & \text { otherwise }\end{cases}\)

(a) Find \(F_{X, Y}(x, y)\), the joint cumulative distribution function (CDF) of \(X\) and \(Y\). A graphical representation is sufficient: probably the best way to do this is to draw the \(x-y\) plane and label different regions with the value of \(F_{X, Y}(x, y)\) in that region.

(b) Let \(Z=X^{2}+Y^{2}\). Find the probability mass function (PMF) of \(Z\).

(c) In Homework #6, you found and sketched \(P_{Y \mid X}\left(y_{j} \mid-1\right)\). Let's use that now to do some estimation of \(Y\) given \(X=-1\); that is, suppose that somebody tells you that \(X=-1\), and we want to come up with a good guess for \(Y\).

- Maximum a Posteriori (MAP) estimator: The MAP estimate of \(Y\) given \(X=-1\) is the value of \(y_{j}\) such that \(P_{Y \mid X}\left(y_{j} \mid-1\right)\) is maximized. Find the MAP estimate of \(Y\) given \(X=-1\).

- Minimum mean squared error (MMSE) estimator: The MMSE estimate of \(Y\) given \(X=-1\) is \(E[Y \mid X=-1]\). Find the MMSE estimate of \(Y\) given \(X=-1\).

Answer 1