Homework 21: Covariance

1. Section 4.6, #5, 11.
2. Suppose that \(X\) and \(Y\) have joint pdf $$ f_{X,Y}(x,y)= \begin{cases} \frac{4}{5}(xy+1), & 0\lt x\lt 1, 0\lt y\lt 1\\ 0, & \text{otherwise} \end{cases} $$ Find Cov\((X,Y)\), the covariance of \(X\) and \(Y\).
3. Let \(X\) be uniformly distributed on \([-1,1]\), and let \(Y=X^2\). Show that Cov\((X,Y)=0\).
4. Let \(X\) and \(Y\) be independent, each with mean 2 and variance 1. Let \(U=3X+2Y\), \(V=2X-3Y\). Find Var \(U\) and Cov\((U,V)\).