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Homework 5

  1. Box Muller: Let $ X_1 \sim \Uc(0,1)$ and $ X_2 \sim
\Uc(0,1)$ (independent). Let

    $\displaystyle Y_1 = \sqrt{-2 \ln X_1} \cos 2\pi X_2 \qquad \qquad
Y_2 = \sqrt{-2 \ln X_1} \sin 2\pi X_2
$

    Show that

    $\displaystyle Y_1\sim \Nc(0,1) \qquad \qquad Y_2 \sim \Nc(0,1).
$

  2. If $ X$ and $ Y$ are independent and $ Y \sim \Uc(0,1)$ , show that $ Z = X+Y$ has the density

    $\displaystyle f_Z(z) = F_x(z) - F_x(z-1).
$

  3. Let $ X \sim \Nc(0,\sigma^2)$ and $ Y \sim \Nc(0,\sigma^2)$ be independent. Let $ Z = X^2 + Y^2$ . Show that

    $\displaystyle f_Z(z) = \frac{1}{2\sigma^2} e^{-z/2\sigma^2} u(z)
$

    Such a random variable is said to be chi-squared $ (\chi^2)$ distributed with two degrees of freedom.
  4. If $ X \in \Uc(0,1)$ and $ Y = -(\ln(X))/\lambda$ , show that $ Y$ is exponentially distributed.
  5. Let $ g(x)$ be a monotone increasing function and let $ Y =
g(X)$ . Show that

    \begin{displaymath}F_{XY}(x,y) =
\begin{cases}
F_X(x) & \text{if } y > g(x) \\
F_Y(y) & \text{if } y < g(x).
\end{cases}\end{displaymath}

  6. Let $ Z = aX + bY$ . Show that

    $\displaystyle f_Z(z) = \frac{1}{\vert a\vert} \int_{-\infty}^\infty
f_{XY}(\frac{z-by}{a},y)  dy.
$

    Hint: Let $ W = Y$ be an auxiliary variable.
  7. Let $ X$ and $ Y$ be independent with

    $\displaystyle f_X(x) = \alpha e^{-\alpha x} u(x) \qquad \qquad f_Y(y) = \beta
e^{-\beta y} u(y).
$

    Determine the density of:
    1. $ Z = X/Y$ .
    2. $ Z = \max(X,Y).$
  8. Show that if $ X,Y$ are i.i.d. $ \Nc(0,\sigma^2)$ , then $ Z = X/Y$ has a Cauchy distribution,

    $\displaystyle f_Z(z) = \frac{1/\pi}{z^2 + 1}.
$

Problems from Grimmet & Stirzaker.
  1. Exercise 4.4.3.
  2. Exercise 4.7.3.
  3. Exercise 4.7.4.
  4. Exercise 4.8.1 (Hint: Laplace transform)
  5. Exercise 4.8.2 (Hint: Laplace transform)

Copyright 2008, Todd Moon. Cite/attribute Resource . admin. (2006, June 13). Homework Assignments. Retrieved January 07, 2011, from Free Online Course Materials — USU OpenCourseWare Web site: http://ocw.usu.edu/Electrical_and_Computer_Engineering/Stochastic_Processes/hw5.html. This work is licensed under a Creative Commons License Creative Commons License