7. Let X,X,,...,X, be a rs from a distribution with mean u and variance o?. Which...
7. Let X, X,,..., X be a rs from a distribution with mean u and variance o”. Which of the following are unbiased estimators of u? If the estimator is biased, compute the bias.
8. Let X, X,, ... , X, be a rs from a distribution with mean u and variance o. Which of п these unbiased estimators has the smallest variance? b) X с) X, а) 4X,-3х,
7. Section 6.4, Exercise 1 Let X. X be a random sample from the U(0,0) distribution, and let , 2X and mx X, be estimators for 0. It is given that the mean and variance of oz are (a) Give an expression for the bias of cach of the two estimators. Are they unbiased? (b) Give an expression for the MSE of cach of the two estimators. (c) Com pute the MSE of each of the two ctrnators for n...
(1 point) A normal distribution with mean and variance o is independently sampled three times, yielding values x1, x2, and X3. Consider the three estimators û = X1 + 5x2 A2 = x - x2 + x3, and Find the expected value of each estimator (type mu for and sigma foro): E) EG) E) = Which estimator(s) are biased and which are unbiased? Estimator : ? Estimator 2: ? Estimators: ? Find the variance of each estimator (type mu for...
7-27. Let X1, X2,..., X, be a random sample of size n from a population with mean u and variance o?. (a) Show that X² is a biased estimator for u?. (b) Find the amount of bias in this estimator. c) What happens to the bias as the sample size n increases?
(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...
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 x and x, be independent random variables with Mean u and variance o2. Suppose that we have two estimators Of u : A @= X1 + X2 2 and ©2 = X, +3X2 2 (a) Are both estimators unbiased estimators of u? (b) What is the variance of each estimator?
Let X1, ..., X10 be a random sample from a population with mean y and variance o?. Consider the following estimators for ji: X1 +...+ X10 3X1 - 2X3 + 3X10 Ô1 = @2 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.
Question 5 15 marks] Let X be a random variable with pdf -{ fx(z) = - 0<r<1 (1) 0 :otherwise, Xa, n>2, be iid. random variables with pdf where 0> 0. Let X. X2.... given by (1) (a) Let Ylog X, where X has pdf given by (1). Show that the pdf of Y is Be- otherwise, (b) Show that the log-likelihood given the X, is = n log0+ (0- 1)log X (0 X) Hence show that the maximum likelihood...