Let X₁, X2,..., Xn be a random sample on the random variable that is uniformly distributed over the interval (1,0) Cit 1:1
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![6. Let X₁, X2, ..., Xn be a random sample on the random variable that is uniformly distributed over
the interval (1,0)
(a) Find the moment estimator, 0* of and the maximum likelihood estimator 0 of 0
(b) Show that 0* is unbiased but Ô is biased
(c) Compute the bias of Ô and find the estimator ♬ that is a function of ô and unbiased.
(d) Find the variance of 0* and 7
(e) Show that both * and are consistent estimators of
(f) Find RE (7,0*) and comment on your results](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa89d6068-0a71-458f-adb0-417a80578597%2Fab6ef8b6-22c0-4569-a9e6-5ea8a2f3f5d7%2Ffnemk7_processed.png&w=3840&q=75)
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- Let X be a random variable with the function, -(x -A) "for x > 2, Let f (x) = { elsewhere. Derive u and show that T = X is biased estimator of 2 . Can you modify T =X in order to get an unbiased estimator of 2.Suppose that Y, Y2,.-.,Y, are independent random variables from a gamma distribution of the parameter (2,6+4). a) Use the method of moment to obtain estimator of B. b) Show that the estimator B, obtained in part a) is unbiased. c) Show that the estimator B is a minimum variance unbiased estimator of ß.Let X1, X2,…. Xn be a random sample from Poisson distribution with parameter , obtain Cramer Rao lower bound for the variance of unbiased estimators of lamda
- a) Let X₁, X₂, X3,... X, be a random sample of size n from population X. Suppose that X-N (0,1) and Y = -√n. i) Show that the standard score of the sample mean X, is equal to Y. ii) Show that the mean and variance of the random variable Y are 0 and 1, respectiv iii) Show using the moment generating function technique that Y is a standard normal random variable.Let Z be nomrally distributed with mean 0 and variance 1. Let Z1 ,.., Zn be an iid random sample, all distributed as Z. Find the moment-generating function of Z2. please show all your work especially when intergrating. do not skip any minor details.Let X₁, X2, X3, X4 be a random sample of size 4 from a population with the following distribution function f(x; 0) = xº e Г(7)В7' if x > 0 elsewhere Where, ß > 0. a) Find the maximum likelihood estimator of the parameter ß. b) Is the maximum likelihood estimator of the parameter ß unbiased or not? c) Find the variance of the maximum likelihood estimator of the parameter ß. d) What is the fisher information in the sample of size n about the parameter ß. e) What is the Cramer-Rao lower bound for the variance of unbiased estimator of the parameter ß? f) Is this maximum likelihood estimator an efficient estimator of ß?
- Let ZZ be a discrete random variable taking one of the four distributions covered in Chapter 10. Suppose you know that Var(Z)=(k+1)E(Z)Var(Z)=(k+1)E(Z) where kk is the last non-zero digit of your student ID number. Determine the distribution of ZZ and find its parameter(s), explaining your argument carefully.Suppose X1, X2, . ,X, be iid random samples from Uniform(a, b), where b > a. ... Find the method of moments estimators for a and b and find the MLES of a and b.Q11. Let X1,X 2,.,X, be a random sample from a Poisson distribution with parameter 2. (a) Find the Cramer-Rao lower bound for the variance of any unbiased estimator of 1. (b) Find the maximum likelihood estimator (m.l.e.) of å. (c) Is this m.l.e. unbiased and does its variance attain the Cramer-Rao lower bound?
- Let Y1, . . . , YN be a random sample from the Normal distribution Yi ∼ N(ln β, s2) where s^2is known.Find the maximum likelihood estimator of b from first principles.Find the Score function, the estimating equation and the information matrix using GLMa) Let X₁, X₂, X3,..., X, be a random sample of size n from population X. Suppose that X~N(0, 1) you and Y = -√n. VR i) Show that the standard score of the sample mean X, is equal to Y. ii) Show that the mean and variance of the random variable Y are 0 and 1, respectively. iii) Show using the moment generating function technique that Y is a standard normal rando... variable. iv) What is the probability that Y² is between 0.02 and 5.02? b) Let X₁, X₂, X, and Y₁, Y₂.... Y be random samples from populations with moment generating functions Mx, (t) = eat+t² and My (t) = (-), respectively. 25 i) Find the sampling distribution of the statistic W = X₁ + 2X₂ X3 + X₁ + X5. ii) What is the value of the sample size n, if PIX(X-X)2> 68.3392] = 0.025? iii) What is the value of the sample size m, if P(|Y-Hy ≥10) <0.04? QUESTION Aa) Let X₁, X₂, X3,..., X₁, be a random sample of size n from population X. Suppose that X~N(0,1) and Y = -√n. i) Show that the standard score of the sample mean X, is equal to Y. ii) Show that the mean and variance of the random variable Y are 0 and 1, respectively. iii) Show using the moment generating function technique that Y is a standard normal random variable. iv) What is the probability that Y² is between 0.02 and 5.02?