1. (Hint: This pmf should look familiar.) Random variables \(X\) and \(Y\) have joint probability mass function \((\mathrm{PMF})\) :

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

1. 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.
2. Let \(Z=X^{2}+Y^{2}\). Find the probability mass function (PMF) of \(Z\).

4. The random variables \(X\) and \(Y\) have joint probability density function \(f_{X, Y}(x, y)\) given by:

\(f_{X, Y}(x, y)= \begin{cases}c x, & 0 \leq x \leq 1, \quad 0 \leq|y| \leq 1-x^{2} \\ 0, & \text { else }\end{cases}\)

1. Find c.
2. Find \(f_{X}(x)\) and \(f_{Y}(y)\), the marginal probability density functions of \(X\) and \(Y\), respectively.
3. Find \(f_{X \mid Y}(x \mid y)\), the conditional probability density function of \(X\) given \(Y\). For your limits (which you should not forget!), put \(y\) between constant bounds and then give the limits for \(x\) in terms of \(y\).
4. Are \(X\) and \(Y\) independent?
5. Consider \(f_{X, Y}(x, y)\) as given above. Suppose that you want to obtain a large value of \(X\) (think of it being money). Suppose that you can choose \(Y=0.25\) or \(Y=0.75\), and then generate \(X\) given that setting of \(Y\). Which \(Y\) do you choose? (There might be multiple good answers. Just be sure to justify yours.)
6. Consider \(f_{X, Y}(x, y)\) as given above. Suppose that you measure \(Y=0.25\). Given that information, what is a good "guess" for the value of \(X\) ? (There might be multiple good answers. Just be sure to justify yours. It might be helpful to look at Problem 1, part (c) for some ideas.)

5. You are again choosing stocks in which to invest, and the set-up is the same as in Homework #5. In particular, the money (in thousands of dollars) made from investing in stocks "Ystock" and "Zstock" are modeled as the random variables \(Y\) and \(Z\), respectively. Assume \(Y\) and \(Z\) are independent with respective probability density functions \(f_{Y}(y)\) and \(f_{Z}(z)\) as shown below:

Suppose you buy Ystock and your sister buys ZStock. What is the probability that you make more money than her; in other words, what is \(P(Y>Z)\) ?

Price: $9.86

Solution: The downloadable solution consists of 5 pages, 486 words and 3 charts.
Deliverable: Word Document