Question 1 [10] Consider a Completely Randomized Design model Yij = μ+ t + εij i = 1,2,...k; j = 1, 2, ... n Show that (i) (ii) E[(2-1)²] = no² i=1 E(MSE) = 0²

Question 1 iid Consider two samples 1, 2, fa with E(X) = µ and finite V(X) iid 91, 92,, yn fy with E(Y) = y and finite V (Y) = 2. Assuming that the two samples are independent, show that -y is consistent estimator for Hr - Hy. ... = o2, and

9. (25 points) Let X and Y be a random variables of the continuous type having the joint pdf f(x, y) = 8xy, %3D (a) [10 points] Find fa(1) and fy(y). (b) [15 points] Find Cov(X,Y) and correlation coefficient.

Consider a random variable Xwith EJ X1=5 and o.’ = 3. Another random variable is given as Y =-8X+10. Find (a) E[X²] (b) E[XY] (c) E[Y²] and (d) 0,.

2. Two dimensional random variable (X, Y) is given by Y / X 1 -2 0,25 0,5 0,25 Find the regression line. [Ans. y = -2x + 2]

If the moment-generating function of X is 2 M(t) = e' + find the mean, variance, and pmf of X. 2t

Please help me

i need help with section b

B) Let X1,X2, .,Xn be a random sample from a N(u, o2) population with both parameters unknown. Consider the two estimators S2 and ô? for o? where S2 is the sample variance, i.e. s2 =E,(X, – X)² and ở² = 'E".,(X1 – X)². [X = =E-, X, is the sample mean]. %3D n-1 Li%3D1 [Hint: a2 (п-1)52 -~x~-1 which has mean (n-1) and variance 2(n-1)] i) Show that S2 is unbiased for o2. Find variance of S2. ii) Find the bias of 62 and the variance of ô2. iii) Show that Mean Square Error (MSE) of ô2 is smaller than MSE of S?. iv) Show that both S2 and ô? are consistent estimators for o?.

iid (6) Let x1, -.. ,X10 d N(µ, o2), where u e R and o > o are unknown parameters. Consider the following four estimators of µ. di = X 10. d2 = 2X1 – X2, X1+X2 d3 = X1+X2+X3 d4 = 2 3 Comparing only these four estimators, point out all the inadmissible estimators. The following `answers" have been proposed. (a) d1, d2, dz are all inadmissible estimators. (b) d1, d2, d4 are all inadmissible estimators. (c) d1, d3, d4 are all inadmissible estimators. (d) Only d3, d4 are inadmissible estimators. (e) None of the above. Joe's argument is, in fact, totally valid. The correct answer is (а) (b) (c) (d) (e) N/A (Select One)