Ruirui Liu

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Numerade tutor for 7 years
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Topics Covered

Mastering Equations and Inequalities: Your Guide to Mathematical Success
Exploring Probability Topics: From Basics to Advanced Strategies
Mastering Integrals: Tips and Tricks for Calculus Success
Integration
Mastering Integration Techniques for Optimal Results

Ruirui's Textbook Answer Videos

08:50
Probability with Applications in Engineering, Science, and Technology

Define a random process $X(t)=A \cos \left(\omega_{0} t+\Theta\right),$ where $A$ and $\Theta$ are independent random variables; $\Theta \sim \operatorname{Unif(}(-\pi \pi, \pi] ;$ and $A$ has mean $\mu_{A}$ and variance $\sigma_{A}^{2} .$ (That is, $X(t)$ models a signal with both phase and amplitude variation.)
(a) Find the mean function of $X(t)$ .
(b) Find the autocorrelation function of $X(t) .$
(c) Is $X(t)$ wide-sense stationary?

Chapter 7: Random Processes
Section 3: Stationary and Wide-Sense Stationary Processes
Ruirui Liu
02:49
Probability with Applications in Engineering, Science, and Technology

Determine whether each of the following functions could potentially be the autocovariance
function of a WSS random process.
(a) $e^{-|\tau+1|}$
(b) $\tau^{2}$
(c) $\operatorname{tri}(\tau),$ defined by tri(\tau) $=1-|\tau|$ for $|\tau| \leq 1$ and 0 otherwise
(d) $\operatorname{sinc}(\tau),$ defined by $\operatorname{sinc}(0)=1$ and $\operatorname{sinc}(\tau)=\sin (\pi \tau) /(\pi \tau)$ for $\tau \neq 0$

Chapter 7: Random Processes
Section 3: Stationary and Wide-Sense Stationary Processes
Ruirui Liu
09:06
Probability with Applications in Engineering, Science, and Technology

Define $X(t)=A t+B,$ where $A$ and $B$ are independent, $A \sim$ Unif $B \sim$ Unif $[-10,10]$
(a) Find the mean function of $X(t)$ .
(b) On the basis of (a), can you determine whether $X(t)$ is WSS? If so, what is you
determination?
(c) Find the variance function of $X(t)$ .
(d) On the basis of (c), can you determine whether $X(t)$ is is $\mathrm{WSS}$ ? If so, what is your
determination?

Chapter 7: Random Processes
Section 3: Stationary and Wide-Sense Stationary Processes
Ruirui Liu
12:53
Probability with Applications in Engineering, Science, and Technology

Let $A(t)$ and $B(t)$ be jointly wide-sense stationary random processes, and define a pair of new
processes by
$X(t)=A(t)+B(t)$
$Y(t)=A(t)-B(t)$
Are $X(t)$ and $Y(t)$ jointly wide-sense stationary?

Chapter 7: Random Processes
Section 3: Stationary and Wide-Sense Stationary Processes
Ruirui Liu
05:33
Probability with Applications in Engineering, Science, and Technology

A wide-sense stationary process $X(t)$ has autocorrelation function $R_{X X}(\tau)=60+125 e^{-|\tau| / 100}$
(a) Does $X(t)$ have any periodic components? How can you tell?
(b) Find the mean square value of $X(t) .$
(c) Find the mean of $X(t),$ if possible.
(d) Find the autocovariance function of $X(t) .$
(e) Find Cov $(X(10), X(15))$ .
(f) Find the standard deviation of $X(t)$

Chapter 7: Random Processes
Section 3: Stationary and Wide-Sense Stationary Processes
Ruirui Liu
10:23
Probability with Applications in Engineering, Science, and Technology

Let $X(t)=A \cdot Y(t),$ where $A$ is a random variable and $Y(t)$ is an ergodic, WSS random process
independent of $A .$
(a) Find the mean and autocorrelation of $X(t)$ in terms of the properties of $A$ and $Y(t) .$ Is $X(t)$
WSS?
(b) Show that the autocovariance function of $X(t)$ is given by $C_{X X}(\tau)=E\left(A^{2}\right) C_{Y Y}(\tau)+\sigma_{A}^{2} \mu_{Y}^{2}$
(c) Find the time average of $X(t) .$ Is $X(t)$ mean ergodic?
(d) Assume $Y(t)$ has no periodic component, so its autocovariance function goes to 0 as
$|\tau| \rightarrow \infty .$ Does the same hold true for the autocovariance function of $X(t) ?$ Why is this
not a violation of property 5 of WSS processes?

Chapter 7: Random Processes
Section 3: Stationary and Wide-Sense Stationary Processes
Ruirui Liu
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