Let x and x, be independent random variables with Mean u and variance o2. Suppose that...
Let X1 and X2 be independent random variables with mean μ and variance σ2. Suppose we have two estimators 1 (1) Are both estimators unbiased estimatros for θ? (2) Which is a better estimator?
Suppose that X and Y are independent random variables with the same unknown mean u. Both X and Y have a variance of 36. Let T = aX + bY be an estimator of u. What condition must a and b satisfy in order that T be an unbiased estimator for ? Is T a normal random variable?
Let X1,..., X10 be a random sample from a population with mean u and variance o2. Consider the following estimators for pe: X1 + ... + X10 ê 3X1 - 2X5 +3X10 10 2 Are these estimators unbiased (i.e. is their expectation equal to u)? A. Both estimators are unbiased. C. Only the second is unbiased. E. Insufficient information. B. Both estimators are biased. D. Only the first is unbiased.
Let X1,X2,...,Xn denote independent and identically distributed random variables with mean µ and variance 2. State whether each of the following statements are true or false, fully justifying your answer. (a) T =(n/n-1)X is a consistent estimator of µ. (b) T = is a consistent estimator of µ (assuming n7). (c) T = is an unbiased estimator of µ. (d) T = X1X2 is an unbiased estimator of µ^2. We were unable to transcribe this imageWe were unable to transcribe...
(1 point) A normal distribution with mean u and variance o2 is independently sampled three times, yielding values X1, X2, and X3 . Consider the three estimators în1 = x1 + 4x2, Û2 = x1 – x2 + x3 , and из şx2 + 3x2 + zxz Find the expected value of each estimator (type mu for u and sigma for o): ECÂ1) = E@2) = ECÂ3) = Which estimator(s) are biased and which are unbiased? Estimator în1: ? Estimator...
Question 3 15 marks] Let X1,..,X be independent identically distributed random variables with pdf common ) = { (#)%2-1/64 0 fx (a;e) 0 where 0 >0 is an unknown parameter X-1. Show that Y ~ T (}, ); (a) Let Y (b) Show that 1 T n =1 is an unbiased estimator of 0-1 ewhere / (0; X) is the log- likeliho od function; (c) Compute U - (d) What functions T (0) have unbiased estimators that attain the relevant...
3. (5 marks) Let U be a random variable which has the continuous uniform distribution on the interval I-1, 1]. Recall that this means the density function fu satisfies for(z-a: a.crwise. 1 u(z), -1ss1, a) Find thc cxpccted valuc and the variancc of U. We now consider estimators for the expected value of U which use a sample of size 2 Let Xi and X2 be independent random variables with the same distribution as U. Let X = (X1 +...
Suppose you have a random sample {X1, X2, X3} of size n = 3. Consider the following three possible estimators for the population mean u and variance o2 Дi 3D (X1+ X2+ X3)/3 Ti2X1/4 X2/2 X3/4 Дз — (Х+ X,+ X3)/4 (a) What is the bias associated with each estimator? (b) What is the variance associated with each estimator? (c) Does the fact that Var(i3) < Var(1) contradict the statement that X is the minimum variance unbiased estimator? Why or...
(a) Let X and Y be independent random variables both with the same mean u +0. Define a new random variable W = ax + by, where a and b are constants. (i) Obtain an expression for E(W). (ii) What constraint is there on the values of a and b so that W is an unbiased estimator of u? Hence write all unbiased versions of W as a formula involving a, X and Y only (and not b). [2]
1. (40) Suppose that X1, X2, .. , Xn, forms an normal distribution with mean /u and variance o2, both unknown: independent and identically distributed sample from 2. 1 f(ru,02) x < 00, -00 < u < 00, o20 - 00 27TO2 (a) Derive the sample variance, S2, for this random sample (b) Derive the maximum likelihood estimator (MLE) of u and o2, denoted fi and o2, respectively (c) Find the MLE of 2 (d) Derive the method of moment...