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- Suppose a time series Yt is given by a random walk Zt and a white noise process Et where Yt = Zt+Et. Find Ps , the coefficients of correlation for Yt.If a random variable X has the moment generating function Mx (t)= 2 - ť Determine the variance of X.Find the PSD of a stationary random process for which Auto correlation is Rx (t)= 6- e alti
- Let X and Y be independent exponential random variables with parameter 1. Find the cumulativedistribution function of Z = X/(X + Y )If it is known that the random process X (t) is a stationary process in a broad sense, could the following autocorrelation function belong to this process? Explain the reason for your answer.Let a stochastic process (Xt) be given as follows: Xt = 3 + 0:5Xt-1 + t, where (t) is white noise with var ( t) = 1. a) What process is (Xt)? b) Explain briefly why this is a \conditional expectation" model, but not a \conditional variance" model. c) Suppose we observed Xt = 2:5, Xt-1 = 1:7. Compute a forecast for Xt+1. d) Suppose we observed Xt = 2:5, Xt-1 = 1:7. Compute a forecast for the variance of the process at time t + 1, that is, for the variance of Xt+1.
- Consider a random process x(t) = Acos(@t+0), where and are constants and 4 is a random variable with zero mean and variance o². Determine whether x(t) is a wide sense stationary process or not.Consider two independent exponential random variables X1 and X2 with parameter lambda=1. LetY1 = X1 Y2 = X1 + X2. Find the MMSE estimate of Y1 using Y2.Calculate the variance of a random variable X whose characteristic function is (1+ e3it)² f(t) = 4
- Suppose Y₁,..., Yn are independent random variables each with the Pareto distribution and E(Y) = (Bo + B1Xi) Is this a generalized linear model with a canonical link? Give reasons for your answer.b) A random variable X follows an exponential distribution, X~Exp(0), with parameter 0 > 0. Find the cumulative distribution function (CDF) of X. i) ii) Show that the moment generating function (MGF) of X is M(t) = iii) Use the MGF in (ii) to find the mean and variance of X.Suppose Y1, Y2, Y3, and Y4 are mutually independent and identically distributed exponential random variables with mean 1. a) Find the moment generating function of the sum T = Y+Y2+Y3+Y4. What is the distribution of T? b) Find the pdf of the maximum order statistic Y(4).