1. The factorization theorem is a powerful tool that we use to identify sufficient statistics.For each of the following cases, find the sufficient statistic. In some cases, no simpli-fication works, and you'll have to say "the whole sample is needed for the sufficientstatistic."(a) X1, X2,..., X, is a random sample from the exponential distribution with meanA. The probability density is e-/[a > 0).(b) X1, X2, ...,X, is a random sample from the Cauchy distribution with median w.(Note that the mean of this distribution doesn't exist). The probabilty density is1for -00 0where -infty < u <∞ and o > 0.(d) X1, X2, ...,X, are independent Poisson random variables with respective meansc, 0 for i = 1,2,...,n, where c1, c2,...,en are n known positive numbers (con-stants). The probability mass functions areI = 1,2,...; 0 > 0, c, > 0for i = 1,2,...,n.

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The factorization theorem is a powerful tool that we use to identify sufficient statistics For each of the following cases, find the sufficient statistic. In some cases, no simpli- fication works, and you'll have to say "the whole sample is needed for the sufficient statistic."

The factorization theorem is a powerful tool that we use to identify sufficient statistics For each of the following cases, find the sufficient statistic. In some cases, no simpli- fication works, and you'll have to say "the whole sample is needed for the sufficient statistic."

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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2-Proportion Test

In a 2-proportion test, two separate proportions are measured at the same time, and their similarity is evaluated. It is also known as a 2-proportion z-test, and a comparison of two different proportions is done in this method.

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Question

1. The factorization theorem is a powerful tool that we use to identify sufficient statistics.
For each of the following cases, find the sufficient statistic. In some cases, no simpli-
fication works, and you'll have to say "the whole sample is needed for the sufficient
statistic."
(a) X1, X2,..., X, is a random sample from the exponential distribution with mean
A. The probability density is e-/[a > 0).
(b) X1, X2, ...,X, is a random sample from the Cauchy distribution with median w.
(Note that the mean of this distribution doesn't exist). The probabilty density is
1
for -00 <I< 0.
*(1+ (x – w)?]
(c) X1, X2, ..., Xn is a random sample from the lognormal distribution with parame-
ters u and o. (Do not think of u and o as the mean and standard deviation here).
The density is
1
(lag
f(r; µ, 0) =
I>0
where -infty < u <∞ and o > 0.
(d) X1, X2, ...,X, are independent Poisson random variables with respective means
c, 0 for i = 1,2,...,n, where c1, c2,...,en are n known positive numbers (con-
stants). The probability mass functions are
I = 1,2,...; 0 > 0, c, > 0
for i = 1,2,...,n.

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