Question

Let X1, X2 be two normal random variables each with population mean µ and population variance σ2. Let σ12 denote the covariance between X1 and X2 and let ¯ X denote the sample mean of X1 and X2. (a) List the condition that needs to be satisfied in order for ¯ X to be an unbiased estimate of µ. (b) [3] As carefully as you can, without skipping steps, show that both X1 and ¯ X are unbiased estimators of µ. (c) [3] Assuming X1 and X2 are independent to each other (σ12 = 0), as carefully as you can, without skipping steps, show that ¯ X is more efficient than X1. (d) [3] Assuming X1 and X2 are dependent to each other with cov(X1,X2) = σ12, as carefully as you can, without skipping steps, to show that V ar( ¯ X) = (σ2 + σ12)/2. (e) [3] Let µ = 5, σ = 10, and σ12 = 10, find P(−1 ≤ ¯ X ≤ 1). (f) [4] Now, instead of normally distributed random variables, let X1, X2,....X1000 be one thousand independently uniformly distributed random variables with the same closed interval [0,5]. Find the density function for Xi, i = 1,2,...,1000. (g) [4] Refer to (f), what is the probability that X1 is greater than 2? (h) [4] Refer to (f), let ¯ X denote the sample mean of X1, X2,....X1000. what is the E( ¯ X) now? (i) [4] Refer to (f), suppose you know the population variance for Xi, V ar(Xi) = 25/3, what is the probability that ¯ X is less or equal to 2? Carefully specify the theorem you used to determine the probability.

Answer 1