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Homework 7

  1. Suppose $ \{X_t, t \geq 0\}$ is a homogeneous Poisson process with parameter $ \lambda$ . Define a random variable $ \tau$ as the time of the first occurrence of an event. Find the p.d.f. and the mean of $ \tau$ .
  2. Suppose { $ X_t, t \in \Rbb\}$ is a w.s.s. random process with autocorrelation function $ R_X(\tau)$ . Show that if $ R_X$ is continuous at $ \tau = 0$ then it is continuous for all $ \tau \in
\Rbb$ . (Hint: Use the Schwartz inequality.)
  3. Under the conditions of problem 2, show that for $ a > 0$ ,

    $\displaystyle P(\vert X_{t+\tau} - X_t\vert \geq a) \leq \frac{2(R_X(0) - R_x(\tau))}{a^2}.
$

  4. Suppose $ A$ and $ B$ are random variables with $ E[A^2] < \infty$ and $ E[B^2] < \infty$ . Define the random processes $ \{X_t, t \in
\Rbb\}$ and $ \{Y_t, t \in \Rbb\}$ by

    $\displaystyle X_t = A + Bt \qquad \qquad Y_t = B + At, \qquad t \in \Rbb.
$

    Find the mean, autocorrelation, and cross correlations of these random processes in terms of the moments of $ A$ and $ B$ .
  5. Let $ p_k(t,s) = X_t - X_s$ , where $ X_t$ is a homoegenous Poisson counting process with rate $ \lambda$ . Show that the differential equation

    $\displaystyle \partiald{}{t} p_k(t,s) = \lambda[p_{k-1}(t,s) - p_k(t,s)]
$

    is solved by

    $\displaystyle p_k(t,s) = \frac{e^{-\lambda(t-s)}(\lambda(t-s))^k}{k!} \qquad
k=0,1,\ldots, \qquad t>s \geq 0.
$

  6. Let $ p_k(t,s) = X_t - X_s$ , where $ X_t$ is an inhomoegenous Poisson counting process with time-varying rate $ \lambda_t$ , where $ \lambda_t$ is a continuous nonnegative function. Show that the differential equation

    $\displaystyle \partiald{}{t} p_k(t,s) = \lambda_t[p_{k-1}(t,s) - p_k(t,s)]
$

    is solved by

    $\displaystyle p_k(t,s) = \frac{e^{-\lambda\int_s^t \lambda_x dx}(\int_s^t
\lambda_x dx)^k}{k!} \qquad
k=0,1,\ldots, \qquad t>s \geq 0.
$

    Also, determine the mean and autocorrelation functions of $ \{X_t, t \in
\Rbb\}$ .

  7. Suppose $ \{X_t, t \in
\Rbb\}$ is a random process with power spectral density

    $\displaystyle S_X(\omega) = \frac{1}{(1+\omega^2)^2}
$

    Find the autocorrelation function of $ X_t$ .
  8. Suppose that $ \omega$ is a random variable with p.d.f. $ f_\omega$ and $ \theta$ is a random variable independent of $ \omega$ uniformly distributed in $ (-\pi,\pi)$ . Define a random process by

    $\displaystyle X_t = a \cos(\omega t + \theta), \qquad t \in \Rbb
$

    where $ a$ is a constant. Find the power spectral density of $ \{X_t\}$ .

  9. Suppose that $ \{X_t, t \in
\Rbb\}$ is a w.s.s., zero-mean, Gaussian random process with autocorrelation function $ R_X(\tau),
\tau \in \Rbb$ and power spectral density $ S_X(\omega), \omega \in
\Rbb$ . Define the random process $ \{Y_t, t \in \Rbb\}$ by $ Y_t =
(X_t)^2, t \in \Rbb$ . Find the mean, autocorrelation, and power spectral density of $ \{Y_t, t \in \Rbb\}$ .
  10. Suppose $ U$ and $ V$ are independent random variables with $ E[U]
= E[V] = 0$ and $ \var(U) = \var(V) = 1$ . Define random processes by

    $\displaystyle X_t = U \cos t + V \sin T \qquad \qquad Y_t = U \sin t + V \cos t,
\qquad t \in \Rbb.
$

    Find the autocorrelation and cross-correlation function s of $ \{X_t, t \in
\Rbb\}$ and $ \{Y_t, t \in \Rbb\}$ . Are $ \{X_t\}$ and $ \{Y_t\}$ jointly wide sense stationary? Are they individually wide sense stationary?


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/hw7.html. This work is licensed under a Creative Commons License Creative Commons License