Let Xt == Yt Y₁ = Zt := = X + nt μ+t-1 Vet with A, ER and where {t} {n} are independent white noise processes with {t} ~ WN(0, 0²), and {n} ~ WN(0,02). Find ex- pressions for the following in terms of model parameters and where possible use notation of Kronecker-Delta functions:

Let Y, and Y, have a bivariate normal distribution. The marginal distribution of Y, is normal with mean u, and variance o, f(y1) = 20í ,-0 < yı < ∞, V2no, Calculate the marginal distribution of f (y2).

I’m having issues understanding the first 2 lines of the proof, variance, E and how it gave the integration. Can you pls write a preliminary explaining those? Thanks.

5. Suppose that Y₁, Y2, Y3 denote a random sample from an exponential distribution with density function 1 f(y) == e 6, y> 0, 00 Y₁ + 2Y₂ Five estimators are proposed for the parameter 0, Ô₁ = Y₁, 0₂: Y₁+Y₂ 03 2 = Ô₁ = Y(1) = " 3 and Ô5 = Y (a) Compute the bias of each estimator and hence identify the unbiased estimators. (b) Find for each of the estimators, the mean square error and hence the standard error. (c) Find the efficiency of 85 relative to each of the other four estimators and comment on your results.

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1. Suppose we model X₁,…, Xn ~ Exp(A) as iid RVs with unknown rate X. Suppose that we observe = 21.79 and n = 100. (a) Conduct a uniformly most powerful test at size a = 0.01 for the test of Ho: A = 1/23 vs H₁: λ=1/22 (b) Recall that the MLE of A is given by  = 1/. Using the MLE as the test statistic show that for the two-sided test of Ho: λ=1/23 vs H₁: λ1/23 at level a = 0.01 the rejection region is of the form n n {x : Σx₁ ≥C or Σxi ≤ D} i=1 i=1 where C and D are constants to be specified. Hint: Use the hinted result in Assignment 3 Qu(1)(f) to help derive the constants. What is the conclusion in this case? Is the Null rejected? (c) Now, using the likelihood ratio test statistic as the test statistic show that the rejection region for the two sided test of Ho: λ=1/23 vs H₁: λ1/23 can be written as (for some constant C) {2: * Exp(-1/27 ) ≤C} 23 (d) For the setting in (c). Derive an expression for -2 log x(x) and evaluate this given the observed data. (e) For the setting in (c). Given…