Q2. Suppose that time series {x} is an AR(1) process X = ax t t-1 t +W, where W W ~N(0, 1)x x x x x x 0 and t t 1 3 2 2 1 3 W is normalized standard white noise with t W~N(0, 1). If we have observed x and x, and we 1 '3' would like to estimate the missing value x2 best linear predictor of x, given x and x 1 3 . Find the
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- Repeat Example 5 when microphone A receives the sound 4 seconds before microphone B.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.Suppose that X is a random variable for which the moment generating function is as follows: MX(t) = e^(2t^2+3t) for −∞ < t < ∞. Find the mean and variance of X. (b) Suppose that X has moment generating function MX(t) =(3e^t/4 + 1/4)^6(i) Find the p.m.f. of X. (ii) Find the mean and variance of X. (c) A person with some finite number of keys wants to open a door. He tries the keys one-by-one independently at random with replacement. How many trails you expect, from him, to open the door? (d) Obtain the form of moment generating function (m.g.f.) for the following p.m.f. – p(x) = ((2^x)(e^-2))/×!, x = 0,1,2, … . Also calculate the mean and variance from m.g.f.
- If the moment-generating function of X is 2 M(t) = e' + find the mean, variance, and pmf of X. 2t3. For a time series with linear trend i.c. Y = Bo + B₁t + Ct, verify the least squares estimators for ₁ on slide 12.If a random variable X has the moment generating function Mx (t)= 2 - ť Determine the variance of X.
- Example 24: If a random variable X has the moment generating function Мx (t) — 2-t' determine the variance of X.(13) Let X b(6,-) find E(5+6x) and distribution function. 3Suppose that n observations are chosen at random from a continuous pdf fY(y). What is the probability that the last observation recorded will be the smallest number in the sample?
- The moment generating function of the random variable X is given by mX(s) = e2e^(t)−2 and the moment generating function of the random variable Y is mY (s) =(3/4et +1/4)10. If it is assumed that the random variables X and Y are independent, findthe following:(a) E(XY)(b) E[(X − Y )2](c) Var(2X − 3Y)Q3: Let X1. ,X, Binomail(1,0). (i) Show that B(1,0) is a member of the exponential class. (ii) Find the minimum variance unbiased estimator (MVUE) of 0.X = max(10,Y) with Y ~ Poisson(lambda=13) a) Calcuate the exact expectation and the variance of X.