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Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018

Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018

18th Edition

ISBN:9780079039897

Author:Carter

Publisher:Carter

Chapter10: Statistics

Section10.1: Measures Of Center

Problem 9PPS

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Question 5
Let X1, X2, ..., X, be a random sample from a distribution with probability density func-
tion
Ox0-1, 0<I < 1;
0,
f(x|0) =
otherwise,
where 0 > 0.
(i) Find the maximum likelihood estimate of 0.
(ii) Find the maximum likelihood estimate of P(X < 1/2).
(iii) Find the Cramer-Rao lower-bound for the variances of unbiased estimators of 0.
(iv) Find a sufficient statistic for 0.
(v) Suppose the prior distribution of e is given by
-ga-le-N0, 0> 0;
fe(0) = { r(a)
0,
otherwise.
Find the posterior distribution of O.
(vi) Based on a = 2, A = 4 and the data x1 = 0.45, x2 = 0.38 and r3 = 0.62, calculate
the (1) posterior mean and (2) posterior mode of the distribution obtained in part (v)
and (3) maximum likelihood estimate of 6.

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