Find the PDF of \(X^3\) for \(X \sim \operatorname{Exp}(\lambda)\).
Let \(U \sim \operatorname{Unif}(-\frac{\pi}{2},\frac{\pi}{2})\). Find the PDF of \(T = \tan(U)\). Be sure to say what the support for \(T\) is.
Prove the Change of Variables Theorem if \(g\) is strictly decreasing instead of increasing.
Suppose that \(X_1, X_2\) are i.i.d. \(\operatorname{Norm}(0,1)\) random variables. Let \[\begin{align*} Y_1 &= X_1 + X_2 \\ Y_2 &= X_1 - X_2 \\ \end{align*}\]
Express the equation above in matrix form. That is, find a matrix \(G\) such that \[\begin{pmatrix}Y_1 \\ Y_2 \end{pmatrix} = G \begin{pmatrix}X_1 \\ X_2 \end{pmatrix}.\]
Is \(G\) invertible? What is its inverse?
Find the Jacobian matrix \[\frac{\partial X}{\partial Y} = \begin{pmatrix} \partial X_1/\partial Y_1 & \partial X_1/\partial Y_2 \\ \partial X_2/\partial Y_1 & \partial X_2/\partial Y_2 \end{pmatrix}.\]
Use the Multivariate Change of Variables Theorem to find the joint density function for \((Y_1,Y_2)\).