Exercise 3.9 The Differencing Filter [V] Suppose that the time series (Xty is given by Xi...
Problem 5.1 (Relation between Gaussian and exponential) Suppose that Xi and X, are i.i.d. N(0,1) (a) Show that Z-X1 + X is exponential with mean 2. b) True or False: Z is independent of Θ-tan ( -i Hint: Use the results from Example 5.4.3, which tells us the joint distribution of V and Θ.
Problem 3.1 Suppose that XI, X2,... Xn is a random sample of size n is to be taken from a Bermoulli distribution for which the value of the parameter θ is unknown, and the prior distribution of θ is a Beta(α,β) distribution. Represent the mean of this prior distribution as μο=α/(α+p). The posterior distribution of θ is Beta =e+ ΣΧ, β.-β+n-ΣΧ.) a) Show that the mean of the posterior distribution is a weighted average of the form where yn and...
Let Xi,... ,Xn be i.i.d with pdf θνθ θ+1 where I(.) denotes the indicator function. (a) Find a 2-dimensional sufficient statistic for the mode (b) Suppose θ is a known constant. Find the MLE for v. (d) Suppose v-1. Find the MLE for and determine its asymptotic distribution. Carefully justify your answer and state any theorems that you use. (e) Suppose1. Find the asymptotic distribution of the MLE estimator of exp[- Let Xi,... ,Xn be i.i.d with pdf θνθ θ+1...
Exercise 2.31 Superposition [] Given two independent weakly stationary time series Xt and Yi) with autocovariance functions x(h) and y (h), show that Zt- Xt +Yt is also weakly stationary, with autocovariance function given by yz(h)-x(h)y(h).
3. Suppose that Xi,.... Xn is a random sample from a uniform distribution over [0,0) That is, 0 elsewhere Also suppose that the prior distribution of θ is a Pareto distribution with density 0 elsewhere where θ0 > 0 and α > 1. (a) Determine (b) Show , θ > max(T1 , . . . ,Zn,%) and hence deduce the posterior density of θ given x, . . . ,Zn is (c) Compute the mean of the posterior distribution and...
Suppose that the position of one particle at time t is given by Xi= 3 sint Yi = 2 cost 0 < t < 211 and the position of a second particle is given by X2 = -3+ cost y2 = 1+ sin t 0<t<21 (a)Graph the paths of both particles. How many points of ntersection are there? (b) Are any of these points of intersection collision points? In other words, are the particles ever at the same place at...
QUESTION 2 Let Xi.. Xn be a random sample from a N (μ, σ 2) distribution, and let S2 and Š-n--S2 be two estimators of σ2. Given: E (S2) σ 2 and V (S2) - ya-X)2 n-l -σ (a) Determine: E S2): (l) V (S2); and (il) MSE (S) (b) Which of s2 and S2 has a larger mean square error? (c) Suppose thatnis an estimator of e based on a random sample of size n. Another equivalent definition of...
Question 2 (50 points]: Continuous-Time Signals Given the following continuous-time signal (t). (t) 5t (a) [4%] What is the fundamental period (i.e., T) and fundamental frequency (ie, wo) of (+)? (b) [8%] Calculate the time average, average power and total energy of x(t). Is x(t) an energy signal? Explain. (c) [8%] Calculate the Fourier series coefficients of (t), i.e., {x}. [Hint: You can make use of the result in Q1(a).] (d) [8%] What is the percentage of power loss if...
You may use the following facts to answer the questions below Fact 1: Suppose that Xi. . . . , X, are independent and X.* GAM (θ.k.) for -1 -1 Fact 2: If Y GAM(0,n aYGAM(ab,n) for any number a >0 1. Suppose that V-GAM(1m) and let lPa θν, where θ > 0. (a) Show that, for any given positive number a, P> a) is an increasing function of (b) What is the probability distribution of W? (c) Would you...
Problem 15-07 (Algorithmic) Refer to the gasoline sales time series data in the given table. 1 Week Sales (1000s of gallons) 17 2 20 21 11 14 22 a. Compute four-week and five-week moving averages for the time series. Round your answers to two decimal places. 4-Week 5-Week Moving Average Moving Average Week Sales 17 9 10 11 12 17 DE 23 21 C C 14 22 b. Compute the MSE for the four-week and five-week moving average forecasts. Round...