Let X1 be the sample mcan of a random sample of size n from a Norinal(jµ, o†) population andX2 be the sample mean of a random sample of the same size n from a Normal(j1, o,) populationand the two samples are independent. Note that the two samples have the same populationImean ji but o + o3.Let i = wX1 +(1 – w)X2,0

Answered: Let X1 be the sample mean of a random…

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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Let X1 be the sample mcan of a random sample of size n from a Norinal(jµ, o†) population and
X2 be the sample mean of a random sample of the same size n from a Normal(j1, o,) population
and the two samples are independent. Note that the two samples have the same population
Imean ji but o + o3.
Let i = wX1 +(1 – w)X2,0 <w<1 be an cstimator for ji. Is îî unbiased for
(a)
L? Explain.
Find the value of w* in [0,1] so that Var(î) is minimized. Hint: take the
(b)
derivative of Var(ſî) with respect to w and set it equal to 0.
(c)
that all population parameters µ, 07,0% are unknown, does it make sense to use îiª as an
estimator for µ? If ycs, explain why. If no, provide a modification of i that has a similar
structure but makes sense as an estimnator for 1.
Let i := w*X1+(1-w*)X2 where w' is the answer from part b. If we assume

Expert Solution

Step 1

A statistic t is said to be an unbiased estimate of a parameter $θ i f E (t) = θ .$

Given

$\hat{μ} = ω {\bar{X}}_{1} + (1 - ω) {\bar{X}}_{2}$

Also

$E ({\bar{X}}_{1}) = E ({\bar{X}}_{2}) = μ$

Consider,

$\begin{array}{rcl} E (\hat{μ}) & = & E (ω {\bar{X}}_{1} + (1 - ω) {\bar{X}}_{2}) \\ = & ω * E ({\bar{X}}_{1}) + (1 - ω) * E ({\bar{X}}_{2}) \\ = & ω μ + (1 - ω) μ \\ = & (ω + 1 - ω) μ \\ = & μ \end{array}$

$∴ \begin{array}{rcl} \hat{μ} \end{array}$ is an unbiased estimator of $μ .$

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