Let {Z} be Gaussian white noise, i.e. {Zt} is a sequence of i.i.d. normal r.v.s each with mean zero and variance 1. Define Xt = [Zt₁ if t is even; (Z²-₁-1)/√√2, if t is odd If {X} and {Y} are uncorrelated stationary sequences, i.e., if X, and Y, are uncorrelated for every r and s, show that {Xt + Yt} is stationary with autocovariance function equal to the sum of the autocovariance functions of {Xt} and {Y}.
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- 3. Let X1, X2, ..., Xn be a random sample from the population N(µz,0²) and Y1, Y2,...,Ym a random sample from the population N(µy, o2), where both means (z and Hy) are assumed known. Note that the two distributions have a common variance of o². (a) If the X's and Y's are independent, show that the MLE for the common variance is: E (Xi - H2)² +E(Y; - Hy)? n+ m (b) Is the MLE found above unbiased for o2?from a random sample of 35 months from January 2006 through December 2020, the mean number of tornadoes per in the united states was about 100. Assume the population standard deviation is 111. Construct a 90% and 95% confidence interval for the population mean.Assume that A, B, and C are subsets of a sample space S with Pr(A) = 0.75, Pr(B) = 0.55, Pr(C) = 0.25. 1. Find Pr(A'), Pr(B'), and Pr(C'). Pr(A') = Pr(B') = %3D Pr(C') = 2. If Pr(A U B) = 0.75., find Pr(A n B). Pr(A n B) = 3. Suppose that we know that B and C are disjoint events. Find Pr[B U C]. Pr[B U C] = 4. Suppose that instead of being disjoint C C B. Find Pr[B U C]. Pr[BU C] =
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