E n Σ(x − x) = (n − 1)σ. - -

2. Suppose the distribution of Y given X = x is a normal random variable with mean x and variance x². And X follows an uniform distribution on (0,1) (a) Find E(XY) (b) Let U = X, V = Y/X. Show that U and V are independent

Q4. Let X1,X 2,.X , be a random sample from an exponential distribution with parameter 0. (a) Find the Cramer-Rao lower bound for the variance of any unbiased estimator of e. (b) Find the maximum likelihood estimator (m.l.e.) of 0. (c) Is this estimator unbiased and does its variance attain the Cramer-Rao lower bound? (d) What is the method of moments estimator of 0?

Let Y, represent the th normal population with unknown mean 4, and unknown variance of for i=1,2. Consider independent random samples, Y₁, Y2. Yin, of size n₁, from the ith population with sample mean Y, and sample variance S² = ₁₁-1(Y₁-₁². j=1 (g) For non-zero constants a's, what is the distribution of U₂ = a₁Y₁-a₂Y₂? State all the relevant parameters of the distribution. (h) Find the standard error of U₂ in part (g), assuming that of = 0² = 0². (i) Discuss how the distribution of Y₁ - ₂ can be used to test the equality of the two population means, #₁ and μ2, when o² = 0 = 0² is known. (j) Define appropriate rejection regions, in terms of Y₁ - Y2, for testing Ho: #₁ = 2 against a two-sided alternative hypothesis at the a level of significance.

1. Random variable X has p.d.f be 6z I> 0 fx(z) = || otherwise and let Y = eX. (b) (i.e. Calculate Var(Y)) Calculate the variance of Y.

If X is a random variable with pdf f(x) = 2x − 2 where x = (1, 2), find the variance of Y = 2X - 3.

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?.