> uppas ?, i-.Tn denote a random Sample of size n fron a Gamma with not-the...
6. Let X1, X2,.. , Xn denote a random sample of size n> 1 from a distribution with pdf f(x; 6) = 6e-8, 0<x< 20, zero elsewhere, and 0 > 0. Le Y = x. (a) Show that Y is a sufficient and complete statistics for . (b) Prove that (n-1)/Y is an unbiased estimator of 0.
I. Let {X n\ be a sequence of random variables wit h E(X,-? for n- 7n exists a C > 0 such that for n 1,2, 3,.. Show that X is cons istent for ?
Problem 5 Let Y1 denote the minimum of a random sample of size n from a distribution that has pdf f(x) e(,0x< o0, zero elsewhere X- n (Y1 0), find the cumulative distribution function (cdf) for Zn = n (Y1 - 0), and Let Zn find the limiting cdf of Zn as n >oo.
1. Consider a random sample of size n from a population with probability density function: х fx(x,0) = e 02 exig for x >0,0 >0. (a) Find the Cramer-Rao lower bound for the variance of an unbiased estimator of (b) Find the methods of moment estimator for @ and verifies that it attains the lower bound
(1) To test Ho: p=0.3; H :p > 0.3, a simple random sample of size n=200 is obtained from a population such that n < 0.05N. (a) If x = 75 and n=200, compute the test statistic zo. (b) Test the hypothesis using (i) the classical approach and (ii) the P-value approach. Assume an a= 0.05 level of significance. (c) What is the conclusion of the hypothesis test?
Let X, X,, ..., X, denote a random sample of size n from a population with pdf (10) = b exp(@m()).0<x<1 where (<O<0. Derive that the likelihood ratio test of H.:0=1 versus H, :0 #1 in terms of T(x) = ŽI (3)
(Stochastic process and probability theory) Let Xn, n > 1, denote a sequence of independent random variables with E(Xn) = p. Consider the sequence of random variables În = n(n-1) {x,x, which is an unbiased estimator of up. Does (a) in f H² ? (6) ûn 4* H?? (c) în + k in mean square? (d) Does the estimator în follow a normal distribution if n + ?
For any two numbers p and q, which of the following must be true? A. Ip+al 2 Ipl + 191 B. Ip-al s lpl - 191 C. Ip-al>o D. Ip.al 2p.q
Let X1, X2, ..., Xn be a random sample from a Gamma( a , ) distribution. That is, f(x;a,0) = loga xa-le-210, 0 < x <co, a>0,0 > 0. Suppose a is known. a. Obtain a method of moments estimator of 0, 0. b. Obtain the maximum likelihood estimator of 0, 0. c. Is O an unbiased estimator for 0 ? Justify your answer. "Hint": E(X) = p. d. Find Var(ë). "Hint": Var(X) = o/n. e. Find MSE(Ô).
Just need to solve Problem 4 4.2.14. Let X denote the mean of a random sample of size 25 from a gamma-type distribution with a = 4 and 3 > 0. Use the Central Limit Theorem to find an approximate 0.954 confidence interval for pl, the mean of the gamma distribution. Hint: Use the random variable (X - 43)/(432/25)/2 = 5X/23 - 10. 21 TL11C1L We were unable to transcribe this image