Math 422 Homework 2

Due Fri, Feb 7

Fri, Jan 24

  1. Show that if \(X, Y\) are independent Exp(1) random variables, then \(X-Y\) has the Laplace distribution. Hint: Recall that the Laplace distribution has MGF \(\frac{1}{1-t^2}\). Also recall that you can find the MGF for \(-Y\) by applying problem 5 from homework 1 to the MGF for \(Y\) since \(-Y\) is just a linear transformation of \(Y\).

Wed, Jan 29

  1. Find the PDF of \(X^3\) for \(X \sim \operatorname{Exp}(\lambda)\).

  2. 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.

  3. Prove the Change of Variables Theorem if \(g\) is strictly decreasing instead of increasing.

Fri, Jan 31

Suppose that \(X_1, X_2\) are i.i.d. \(\operatorname{Norm}(0,1)\) random variables. Let \[\begin{align*} Y_1 &= X_1 + X_2 \\ Y_2 &= X_1 - X_2 \\ \end{align*}\]

  1. Express the equation above in matrix form. That is, find a matrix \(G\) such that \[\begin{pmatrix}Y_1 \\ Y_2 \end{pmatrix} = G \begin{pmatrix}X_1 \\ X_2 \end{pmatrix}.\]

  2. Is \(G\) invertible? What is its inverse?

  3. Find the Jacobian matrix \[\frac{\partial X}{\partial Y} = \begin{pmatrix} \partial X_1/\partial Y_1 & \partial X_1/\partial Y_2 \\ \partial X_2/\partial Y_1 & \partial X_2/\partial Y_2 \end{pmatrix}.\]

  4. Use the Multivariate Change of Variables Theorem to find the joint density function for \((Y_1,Y_2)\).

Mon, Feb 3

  1. Any two linearly independent vectors \(a, b \in \mathbb{R}^n\) form a parallelogram with vertices 0, \(a\), \(b\), and \(a+b\). Prove that if this parallelogram is a rhombus (i.e., all four sides have the same length), then the diagonals are orthogonal.
0 a b a + b