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Homework 7

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 $\tau$ .
Suppose { $X_t, t \in \Rbb\}$ is a w.s.s. random process with autocorrelation function $R_X(\tau)$ . Show that if is continuous at $\tau = 0$ then it is continuous for all $\tau \in \Rbb$ . (Hint: Use the Schwartz inequality.)
Under the conditions of problem 2, show that for 0$" align="middle" border="0" height="34" width="49" /> ,

$\displaystyle P(\vert X_{t+\tau} - X_t\vert \geq a) \leq \frac{2(R_X(0) - R_x(\tau))}{a^2}.$
Suppose and are random variables with $E[A^2] < \infty$ < \infty$" align="middle" border="0" height="41" width="98" /> and $E[B^2] < \infty$ < \infty$" align="middle" border="0" height="41" width="99" /> . 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 and .
Let , where 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

$\displaystyle p_k(t,s) = \frac{e^{-\lambda(t-s)}(\lambda(t-s))^k}{k!} \qquad k=0,1,\ldots, \qquad t>s \geq 0.$ s \geq 0. $" align="middle" border="0" height="68" width="489" />
Let , where 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

$\displaystyle p_k(t,s) = \frac{e^{-\lambda\int_s^t \lambda_x dx}(\int_s^t \lambda_x dx)^k}{k!} \qquad k=0,1,\ldots, \qquad t>s \geq 0.$ s \geq 0. $" align="middle" border="0" height="79" width="508" />

Also, determine the mean and autocorrelation functions of $\{X_t, t \in \Rbb\}$ .
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 .
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 is a constant. Find the power spectral density of $\{X_t\}$ .
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 $\{Y_t, t \in \Rbb\}$ .
Suppose and are independent random variables with 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 November 24, 2009, 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.

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	Info Homework Assignments Document Actions Homework 7 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 $\tau$ . Suppose { $X_t, t \in \Rbb\}$ is a w.s.s. random process with autocorrelation function $R_X(\tau)$ . Show that if is continuous at $\tau = 0$ then it is continuous for all $\tau \in \Rbb$ . (Hint: Use the Schwartz inequality.) Under the conditions of problem 2, show that for 0$" align="middle" border="0" height="34" width="49" /> , $\displaystyle P(\vert X_{t+\tau} - X_t\vert \geq a) \leq \frac{2(R_X(0) - R_x(\tau))}{a^2}.$ Suppose and are random variables with $E[A^2] < \infty$ < \infty$" align="middle" border="0" height="41" width="98" /> and $E[B^2] < \infty$ < \infty$" align="middle" border="0" height="41" width="99" /> . 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 and . Let , where 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 $\displaystyle p_k(t,s) = \frac{e^{-\lambda(t-s)}(\lambda(t-s))^k}{k!} \qquad k=0,1,\ldots, \qquad t>s \geq 0.$ s \geq 0. $" align="middle" border="0" height="68" width="489" /> Let , where 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 $\displaystyle p_k(t,s) = \frac{e^{-\lambda\int_s^t \lambda_x dx}(\int_s^t \lambda_x dx)^k}{k!} \qquad k=0,1,\ldots, \qquad t>s \geq 0.$ s \geq 0. $" align="middle" border="0" height="79" width="508" /> Also, determine the mean and autocorrelation functions of $\{X_t, t \in \Rbb\}$ . 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 . 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 is a constant. Find the power spectral density of $\{X_t\}$ . 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 $\{Y_t, t \in \Rbb\}$ . Suppose and are independent random variables with 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 November 24, 2009, 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.	Reuse Course Download this course

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