Onsider the process Y, = Y + Σ|e, where Yo ~ (μ, σ2) and the e's are 0-mean, a stationary process...
3. Suppose X1,X2, are independent identically distributed random variables with mean μ and variance σ2. Let So = 0 and for n > 0 let Sn denote the partial sumi Let Fn denote the information contained in X1, ,Xn. (1) Verify that Sn nu is a martingale. (2) Assume that μ 0, verify that Sn-nơ2 is a martingale. 3. Suppose X1,X2, are independent identically distributed random variables with mean μ and variance σ2. Let So = 0 and for n...
Ifx, are normally distributed random variables with mean μ and variance σ2, then: and σ are the maximum likelihood estimators ofμ and σ2, respectively. Are the MLEs unbiased for their respective parameters?
Y2 Let Yİ'ý, variance σ. We showed in class that minimally sufficient estimators. Does this imply that y and S2 are MVUE estimators of μ and X, be independent and identically distributed random variables with mean μ and (Ση! YǐΣι Y?) is sufficient for (μ, σ2). These are also σ2 respectively? Explain why or why not.
1. Let Y.Y2, ,y, be independent and identically distributed N(μ, σ2) random variables. Show that, where d() denotes the cumulative distribution function of standard normal [You need to show both the equalities]
Consider the process Y.-μ + et-o, et-1-912 et-12, where {ed denotes a white-noise process with mean 0 and variance σ? > 0. Assume that et ls independent of Yt-1, Yt-2, Find the autocorrelation function for (Yt).
QUESTION4 (a) Let e be a zero-mean, unit-variance white noise process. Consider a process that begins at time t = 0 and is defined recursively as follows. Let Y0 = ceo and Y1-CgY0-ei. Then let Y,-φ1Yt-it wt-1-et for t > ï as in an AR(2) process. Show that the process mean, E(Y.), is zero. (b) Suppose that (a is generated according to }.-10 e,-tet-+扣-1 with e,-N(0.) 0 Find the mean and covariance functions for (Y). Is (Y) stationary? Justify your...
just part a plz thank u! Page 4 Marks Suppose Z(t) Y., where X(t) is the Poisson process with rate θ If μ = E[h] and σ2-Yar determine the mean and variance of Z(t) a. pil are the common mean and variance for y,y , then b. fVis Uniform distribution on interval (0, 1], then determine the mean and variance of XCV) 2 Page 4 Marks Suppose Z(t) Y., where X(t) is the Poisson process with rate θ If μ...
QUESTION 3 (a) Consider the ARMA (1, 1) process -Bat-1-where o and θ are model parame- are independent and identically distributed random variables with mean 0 z, oz,-1 ters, and a1, a2, and variance σ (i) Show that the variance of the process is γ,- (ii) Using (i) or otherwise, show that the autocorrelation function (ACF) of the process is: ifk=0. (b) Let Y be an AR(2) process of the special form Y-2Y-2e (i) Find the range of values of...
Let Xi, x,, ,X, be independent random variables with mean and variance σ . Let Y1-Y2, , Y, be independent random variables with mhean μ and variance a) Compute the expected value of W b) For what value of a is the variance of W a minimum? σ: Let W-aX + (1-a) Y, where 0 < a < 1. Let Xi, x,, ,X, be independent random variables with mean and variance σ . Let Y1-Y2, , Y, be independent random...
Problem 2. Let X be a random variable with mean 0 and variance σ2. Define the process Yt-(-1) Compute the mean and covariance function of the process {Yt). Is this process stationary?