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Homework 4

  1. Suppose $ \Xbf \sim \Nc(\mubf,\Sigma)$ .
    1. Show that $ E[\Xbf] = \mubf$ and $ \cov(\Xbf,\Xbf) = \Sigma$ .
    2. Show that $ A \Xbf + \bbf \sim \Nc(A \mubf + \bbf, A \Sigma
A^T)$ .
    3. Suppose $ \Sigma > 0$ and write $ \Sigma = CC^T$ . Show that $ C^{-1}(\Xbf - \mubf) \sim \Nc(\zerobf,I)$ .
  2. Suppose you have a random number generator which is capable of generating random numbers distributed as $ X \sim \Nc(0,1)$ . Describe how to generate random vectors $ \Ybf \sim \Nc(\mubf,\Sigma)$ .

  3. Suppose $ X$ and $ Y$ are r.v.s. Show that $ E[(X - h(Y))^2]$ is minimized over all functions $ h$ when $ h$ is the function

    $\displaystyle h(y) = E[X\vert Y=y].
$

    Assume $ E[X^2] < \infty$ .

  4. Let $ (X,Y) \sim \Nc(\mu_x,\mu_y,\sigma_x^2,\sigma_y^2,\rho)$ . Let $ \Xbf = \left[\begin{smallmatrix}X \\
Y\end{smallmatrix}\right] \sim \Nc(\mubf, \Sigma)$ , where

    $\displaystyle \Sigma = \begin{bmatrix}s_{11} & s_{12}  s_{21} & s_{22}
\end{bmatrix}.
$

    Determine the relationship between $ \mu_x,\mu_y,\sigma_x^2,\sigma_y^2,\rho$ and $ \mubf$ and $ \Sigma$ .

  5. Suppose $ \Xbf = \left[\begin{smallmatrix}X_1  X_2  X_3\end{smallmatrix}\right]
\sim \Nc(\mubf,\Sigma)$ where

    $\displaystyle \mubf = \begin{bmatrix}1  2  3
\end{bmatrix}\qquad \qquad
\Sigma = \begin{bmatrix}4 & 2 & 1  2 & 6 & 3  1 & 3 & 8
\end{bmatrix}$

    1. The value $ X_1 = 1.5$ is measured. Determine the best estimate for $ (X_2,X_3)$ .
    2. In a separate problem, the values $ X_2 = 1$ and $ X_3 = 5$ are measured. Determine the best estimate of $ X_1$ .
    3. Determine a random vector $ \Ybf$ which is a whitened version of $ \Xbf$ .
Problems from Grimmet & Stirzaker
  1. Ex 3.7.5. What is requested is $ E[T-t\vert T>t]$ , i.e., the mean subsequent lifetime given that the machine is still running after $ t$ days. Then use the hint from the book. Note that in (a), $ P(T>t) = \frac{1}{N+1}(N-t)$ .
  2. Ex 3.7.7. Hint: Show that $ P($ robot faulty $ \vert$ fault not detected $ ) = \frac{\phi(1-\delta)}{1-\phi\delta} \defeq \pi$ . Hence argue that the number of faulty passed robots, given $ Y$ , is distributed as $ \Bc(n-Y,\pi)$ , which has mean $ (n-Y)\pi$ . Hence show that $ E[X\vert Y] = Y + (n-Y)\pi$ .
  3. Ex 4.1.1(a)
  4. Ex 4.1.2
  5. Ex 4.2.1. Hint: think geometric r.v.
  6. Ex 4.2.2. Hint: $ P($ max $ (X,Y) \leq v) = P(X \leq v, Y \leq v)$ .
  7. Ex 4.4.1. Hint: integrate by parts.
  8. Ex 4.6.4(b)

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/hw4.html. This work is licensed under a Creative Commons License Creative Commons License