(b) Let (etez be standard Gaussian, N(0, 1), white noise, and define a first-order au- toregressi...
3. Let Zt) be a Gaussian white noise, that is, a sequence of i.i.d. normal r.v.s each with mean zero and variance 1. Let Y% (a) Using R generate 300 observations of the Gaussian white noise Z. Plot the series and its acf. (b) Using R, plot 300 observations of the series Y -Z. Plot its acf. c) Analyze graphs from (a) and (b). Can you see a difference between the plots of graphs of time series Z and Y?...
Let Xn, -inf to +inf be a discrete-time zero-mean white noise process, i.e., μx[n] = 0, Rx[n] =δ[n]. The process is filtered using an LTI system with impulse response h[n] =αδ[n] + βδ[n−1]. Find α and β such that the output process Yn has autocorrelation function Ry[n] =δ[n+1] + 2δ[n] +δ[n−1]. 5) (3 points) Let Xn, -o0 K n oo, be a discrete-time zero-mean white noise process, i.e, ,1z[n]-(), Rx [n] S[n]. The process is filtered using an LTI system...
7. Let Z be Gaussian white noise, i.e. Z is a sequence of i.i.d. normal r.v.s each with mean zero and variance 1. Define Zt, t(-1- 1)/v2, if t is odd Show that Xis WN(0,1) (that is, variables Xt and Xt+k,k2 1, are uncorrelated with mean zero and variance 1) but that Xt and Xi-i are not i.i.d 7. Let Z be Gaussian white noise, i.e. Z is a sequence of i.i.d. normal r.v.s each with mean zero and variance...
5.57 Let np(t) be a zero-mean white Gaussian noise with the power spectral density 20 let this noise be passed through an ideal bandpass filter with the bandwidth 2W centered at the frequency fe. Denote the output process by nt). 1. Assuming fo fe, find the power content of the in-phase and quadrature components of n(t). We were unable to transcribe this image 5.57 Let np(t) be a zero-mean white Gaussian noise with the power spectral density 20 let this...
Let {et} denote a white noise process from a normal distribution with E[et] = 0, Var(et) = σe2 and Cov(et, es) = 0 for t ≠ s. Define a new time series {Yt} by Yt = et + 0.6 et -- 1 – 0.4 et – 2 + 0.2 et – 3. 1. Find E(Yt ) and Var(Yt ). 2. Find Cov(Yt , Yt – k) for k = 1, 2, ...
1. [30 pts! Let Yǐ follow a moving average process of order 1 (ie, MA(1): where e is a white noise process with N(0,1). Suppose that you estimate the model using STATA. You obtain ê-1, ê-0.5 and ớ2-1. You also know e,-2 and E1-1-3. (a) Obtain the unconditional mean and variance of Y (b) Obtain Cor(Y, Yi-1). (c) Obtain the autocorrelation of order 1 for Y 1. [30 pts! Let Yǐ follow a moving average process of order 1 (ie,...
2. Let (et) be a zero mean white noise process with variance 1. Suppose that the observed process is h ft + Xt where β is an unknown constant, and Xt-et- Explain why {X.) is stationary. Find its mean function μχ and autocorrelation function p for lk0,1,.. a. b. Show that {Yt3 is not stationary. C. Explain why w. = ▽h = h-K-1 is stationary. d. Calculate Var(Yt) Vt and Var(W) Vt . (Recall: Var(X+c)-Var(X) when c is a constant.)...
Exercise 2. Let Xn, n EN, be a Bernoulli process uith parameter p = 1/2. Define N = min(n > 1:X,メ } For any n 2 1, define Yn = XN4n-2. Show that P(Yn = 1) = 1/2, but Yn, n E N is not a Bernoulli process Exercise 2. Let Xn, n EN, be a Bernoulli process uith parameter p = 1/2. Define N = min(n > 1:X,メ } For any n 2 1, define Yn = XN4n-2. Show...
2. i) Let B be a random variable with the Binomial (n, p) distribution, so that Write down the likelihood function L(p) for m independent observations xi,...,Inm 2 marks 6 marks ili) Compute the bias and the mean squared error of the corresponding maximum likeli- from B. Int ii) Show that the maximum likelihood estimate for pis-Σ.ri. mn [7 marks] hood estimator of p. iv) Let X be a continuous random variable with density function for x > 0, and...
Let Z1, Z2, . . . be a sequence of independent standard normal random variables. Define X0 = 0 and Xn+1 = (nXn + (Zn+1))/ (n + 1) , n = 0, 1, 2, . . . . The stochastic process {Xn, n = 0, 1, 2, } is a Markov chain, but with a continuous state space. (a) Find E(Xn) and Var(Xn). (b) Give probability distribution of Xn. (c) Find limn→∞ P(Xn > epsilon) for any epsilon > 0.