Please see attached mathematical statistics question below. How to show the unbiased estimator of theta has variance2theta^2/n. Please do not reject question if you do not know how to solve this problem. Allow another expert a chance to answer this question. **Unbiased Estimator Problem**Let \( X_1, \ldots, X_n \) denote a random sample from a normal distribution with mean zero and variance \( \theta > 0 \). Show that \( \frac{1}{n} \sum_{i=1}^{n} X_i^2 \) is an unbiased estimator of \( \theta \) and has variance \( 2\theta^2 / n \).

It is a fact of statistics . It is widely used in probability .X1,X2,......,Xn ~ iid N(0,θ)Then , E(Xi)=0 and V(Xi)=θ , i=1(1)nE(1n∑i=1nXi2)= E(X12) = V(X1)+E2(X1) = θ Hence , 1n∑i=1nXi2 is an unbiased estimator of θ V( 1n∑i=1nXi2 ) =V( X12)n = E(X14) - E2(X12) Now , to find E(X14) , we proceed :Lemma : If Y be a normal variable with mean μ and variance σ2, then E(Y4) =3σ4Here , X1θ ~ N(0,1)E[(X1θ)4] =3 , E(X14) =3θ2V( 1n∑i=1nXi2 ) =V( X12)n = E(X14) - E2(X12) n =3θ2-θ2n =2θ2n

Answered: Let X1,..., Xn denote a random sample…

Math

Probability

Let X1,..., Xn denote a random sample from a normal distribution with mean zero and variance 0 > 0. Show that ", X? is an unbiased estimator of 0 and has variance 26 /n.

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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