(a) Let {X(t), te R} be a continuous-time random process, defined as X(t) = A cos (2t + $), where A U(0, 1) and ~ U(0, 2π) are two independent random variables. (i) Find the mean function µx(t). (ii) Find the correlation function Rx (t1, t2). (iii) Is X(t) a widely stationary stochastic process? (b) Let X(t) be a complex-valued random process defined as Y(A). Ani(wt+$)
(a) Let {X(t), te R} be a continuous-time random process, defined as X(t) = A cos (2t + $), where A U(0, 1) and ~ U(0, 2π) are two independent random variables. (i) Find the mean function µx(t). (ii) Find the correlation function Rx (t1, t2). (iii) Is X(t) a widely stationary stochastic process? (b) Let X(t) be a complex-valued random process defined as Y(A). Ani(wt+$)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
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(a) Let {X(t), t = R} be a continuous-time random process, defined as
X(t) = A cos (2t + $),
where A~ U(0, 1) and ~ U(0, 2) are two independent random variables.
(i) Find the mean function #x(t).
(ii) Find the correlation function Rx (t1, t2).
(iii) Is X(t) a widely stationary stochastic process?
(b) Let X(t) be a complex-valued random process defined as
X(t) = Aej(wt+),
where j = √-1, ~ U(0, 2π), and A is a random variable independent of with
E[A] = μ and Var(A) = o².
(i) Find the mean function of X(t), ux(t).
(ii) Find the autocorrelation function of X(t), Rx (t1, t2).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fea0b1f8a-63ea-4896-a7a1-0b6830f20fb1%2F3ca3b761-3b6a-42a1-a6dc-07a171b2053e%2Fcfcoaih_processed.png&w=3840&q=75)
Transcribed Image Text:3
(a) Let {X(t), t = R} be a continuous-time random process, defined as
X(t) = A cos (2t + $),
where A~ U(0, 1) and ~ U(0, 2) are two independent random variables.
(i) Find the mean function #x(t).
(ii) Find the correlation function Rx (t1, t2).
(iii) Is X(t) a widely stationary stochastic process?
(b) Let X(t) be a complex-valued random process defined as
X(t) = Aej(wt+),
where j = √-1, ~ U(0, 2π), and A is a random variable independent of with
E[A] = μ and Var(A) = o².
(i) Find the mean function of X(t), ux(t).
(ii) Find the autocorrelation function of X(t), Rx (t1, t2).
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