2. Let X and Y be random variables. What is Cov[X,E[Y]] ? What about Cov[E[X],Y] ? 3. Suppose \( X \perp Y \). Derive what Cov[X,Y] is Hint: Although I did not give you the joint distribution of (X,Y), you can still calculate the value of Cov[X,Y]. 4. In class, we discussed how Cov[X,Y]=0 does not imply \( X \perp Y \). The example we considered was Y=X2. Suppose X is uniformly distributed over the interval [1,1], i.e. fX(x)={0.50forx[1,1]otherwise Show that Cov[X,Y] is indeed 0 . 5. In the previous question, the distribution of X was symmetric around 0 . Consider different distributions of X. Do you think it will always be the case that Cov[X,Y]=0 when Y=X2 ? Provide an intuitive explanation for your thoughts, no need for derivations. 6. Is \( X \perp Y \) a "stronger" condition than Cov[X,Y]=0 ? Or is it the other way around? Or are they equally strong? See the hint on the next page. Sometimes, we are interested in the distribution of averages of random variables, e.g., the coin toss question from Problem Set 1. So let X1 and X2 be random variables (not necessarily coin tosses), and let X(X1+X2)/2. 7. Write the expression for Var[X]. Hint: Try to reorder terms to get XiE[Xi] wherever you can. Once you get that difference, treat it like a single term. 8. If I gave you only the marginal distributions of X1 and X2, what is the weakest condition you need to calculate Var[X] ? Takeaway: Independence between random variables pertains to the entire joint distribution. Covariance is just a summary of the joint distribution. Restrictions on the dependence between random variables - whether it is through their joint distribution or covariance-can be very helpful. The final question demonstrates how useful these restrictions can be for learning the distribution of an average of random variables, which is something we often care about