Question: Suppose X and Y have a joint PDF

\[f\left( x,y \right)=\frac{1}{8\pi }\left\{ \begin{aligned} & 4-{{x}^{2}}-{{y}^{2}},\,\,\,\,\,{{x}^{2}}+{{y}^{2}}\le 4 \\ & \,\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{otherwise} \\ \end{aligned} \right.\]

(a) Calculate P(X² + Y² ≤ 1).

(b) Calculate the marginal PDF for X alone.

(c) What is the covariance between X and Y?

(d) Find an event depending on X alone whose probability depends on Y. Use this to show that X is not independent of Y.

(e) Write the joint PDF for U = X² and V = Y² .

(f) Calculate the covariance between X² and Y² . It may be easier to do this without using part(e). Use this to show, again, that X and Y are not independent.