d)) A random process is defined as X(t) = μ + N(t), where N(t) is white noise with autocovariance function Cxx (t₁, t₂) = 0(t₁)8(t₁- t₂), where Ø(t) is abounded functionof t and 8 is the unit impulse function, prove that {X(t)} is a mean ergodic process.DARTD2035

Given: A random process is defined as , where is white…

Answered: (d)) A random process is defined as…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Suppose Xn is an IID Gaussian process, withµX[n]=1, and σ2 X[n]=1Now, another stochastic process Yn = Xn − Xn−1. Please find:(a) The mean µY (n).(b) The variance σ2Y (n).(c) The auto-correlation RY (n, k)
Consider a random process X(t) = A cos o t where a' is a constant and A is a random variable uniformly distributed over (0, 1). Find the auto correlation and covariance of X(t).
A WSS process X(t) 3 Sxx (w) = 507 d(w) + 1+ with a mean value 5 and power spectrum is applied to a network with impulse response (0) 2 h(t) = 4 e4|t| Find H() of the network.
2. Let X and Y be jointly continuous random variables with joint PDF x + cy2 0, OSXS1,0Sys1 elsewhere a) Find the constant c Find the marginal PDF's fy(x) and fy(y) c) Find P(OSXS1/2,0SYS1/2) b)
A WSS process X(t) 3 Sxx (w) = 50r d(w) + 1+ with a mean value 5 and power spectrum is applied to a network with impulse response (0) 2 h(t) = 4 e4|t)| i) Find H(w) of the network.
2 Let N(t) be a zero mean wide sense stationary noise process for which RNN()= No 8(t) where No > 0 is a finite constant. Determine whether N(t) is mean 2 ergodic.
Find out whether each of the following processes {Xt} is weakly stationary or nonstationary. For the weakly stationary processes, find the mean and autocovariance functions. Consider in each question that {et} ~ WN(0, 0²). and X₁ = Xt-1 for t ≥ 2. X₁ = { 1 with probability p otherwise 0
Suppose P(X = 2) = 1/2, P(X = 5) = 1/3 and P(X = 7) = 1/6(a) Compute the probability generating function rX(t)(b) Verify that r'X(0) = P(X = 1) and rX”(0) = 2P(X = 2)(c) Compute the moment generating function of X, mX(s)(d) Verify that m'X(0) = E(X) and mX”(0) = E(X2)
7. Suppose that X is a continuous random variable with pdf Se-(x+2), 2), if-2 < x < 0 f(x)= = lo, elsewhere a) Find the moment generating function of X. b) Use the MGF of (a) to find E(X) and E(X²). 8.
Let X.,X,"(4,0²). Consider the fotlowing estimators of u A, =(X+ Xs + X, + X10) A = (X2+ X, +Xp). Then (a) A, is more efficient than Az (b iz is more efficient than i (e) Can't deside (d) None
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(d)) A random process is defined as X(t) = µ + N(t), where N(t) is white noise with auto covariance function Cxx (t₁, t₂) = Ø(t₁)8(t₁ - t₂), where Ø(t) is abounded function of t and 8 is the unit impulse function, prove that {X(t)} is a mean ergodic process.