situation. Suppose X1,..., Xn d N(μ₁, σ²) and Y₁, . 22d ~ , Xm 2 N(μ2, 02) are all independent, and we'd like to test Ho M1 = μ2. Let A(0; D) denote the likelihood ratio-here, evaluated on the set o D= {X1, Xn, Y₁,..., Ym}. = {(μ1, M2) μ1 = M2} CR2 and for (a) Briefly state the answers to the following questions: (i) What are the MLEs of μ1, μ2, σ² under the current model assumption? (ii) What are the MLEs of μ1, 2, σ² while being restricted to the set no? (b) Show that 2 log A(0; D) = (n+m) log 1+ nm(XY)²] (n+m)S² where X Ŷ = Y₁ + ··· + Ym n m n m and S² = Σ(X; – X)² + Σ(¥¿ − Ÿ)². = [(X;- i=1 j=1 [Hint: You can use the fact that, given a set of numbers {1,..., zn} and their average z = identity ±1(zi — a)² = Σ²²±1(²i − z)² + n(z − a)² holds for any a. - (21++zn)/n, the

Let X1, , Xµ be iid with population density (1 0) I>0, Sx(x) = %3D otherwise. Here 0 is an unkown population parameter. 0 has an Exponential(1) distribution. Find the method of moment estimator for 0. Let's call this 6. Is ô unbiased for 0 ? Explain with precise computation. Show that X Find the maximum likelihood estimator for 0. Let's call this 62. Is ô2 unbiascd for 0 ? Explain with precise computation.

Let X1,..., X10 be a random sample of size 10 from a N(u, o²) population. Suppose 10 Y =(X; - 4)? i=1 Find the probability that the random interval Y Y 20.5' 3.25 includes the point o?.

(4) Consider n i.i.d. samples of X ~ N(µ,0²). Find the maximum likelihood estimate of o?.