/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Q3E Question:聽For the conditions of... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Q3E (page 527)

Question:For the conditions of Exercise 2, find an unbiased estimator of \({\left( {{\bf{E}}\left( {\bf{X}} \right)} \right)^{\bf{2}}}\). Hint: \({\left( {{\bf{E}}\left( {\bf{X}} \right)} \right)^{\bf{2}}}{\bf{ = E}}\left( {{{\bf{X}}^{\bf{2}}}} \right){\bf{ - Var}}\left( {\bf{X}} \right)\)

Short Answer

Expert verified

\(\frac{1}{n}{\sum\limits_{i = 1}^n {{X_i}} ^2} - \frac{1}{{n - 1}}\sum\limits_{i = 1}^n {{{\left( {X - \overline x } \right)}^2}} \)

Step by step solution

01

Given information

For a sample of size n, from a population with mean \(\mu \)and finite variance \({\sigma ^2}\) .

The equation of variance is given as:

\(Var\left( X \right) = E\left( {{X^2}} \right) - {\left( {E\left( X \right)} \right)^2} \ldots \left( 1 \right)\)

02

Rewriting the equationand finding the terms

By rewriting the equation (1)

\({\left( {E\left( X \right)} \right)^2} = E\left( {{X^2}} \right) - Var\left( X \right) \ldots \left( 2 \right)\)

Since we are considering sample estimates, we will substitute \(\mu \) with \(\overline x \).

Now,

\(E\left( {{X^2}} \right) = \frac{1}{n}{\sum\limits_{i = 1}^n {{X_i}} ^2} \ldots \left( 3 \right)\)

We know that,

\(Var\left( X \right) = E{\left( {X - \mu } \right)^2}\)

By replacing \(\mu \) with \(\overline x \)

\(\begin{aligned}{c}Var\left( X \right) &= E{\left( {X - \overline x } \right)^2}\\ &= \frac{1}{{n - 1}}\sum\limits_{i = 1}^n {{{\left( {X - \overline x } \right)}^2}} \ldots \left( 4 \right)\end{aligned}\)

03

Replacing the terms in the equation

By replacing (3) and (4) in (2), we get,

\(\begin{aligned}{}{\left( {E\left( X \right)} \right)^2} &= E\left( {{X^2}} \right) - Var\left( X \right)\\ &= \frac{1}{n}{\sum\limits_{i = 1}^n {{X_i}} ^2} - \frac{1}{{n - 1}}\sum\limits_{i = 1}^n {{{\left( {X - \overline x } \right)}^2}} \end{aligned}\)

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Consider the conditions of Exercise 10 again. Suppose also that it is found in a random sample of size n = 10 \(\overline {{{\bf{x}}_{\bf{n}}}} {\bf{ = 1}}\,\,{\bf{and}}\,\,{{\bf{s}}_{\bf{n}}}^{\bf{2}}{\bf{ = 8}}\) . Find the shortest possible interval so that the posterior probability \({\bf{\mu }}\) lies in the interval is 0.95.

Consider again the conditions of Exercise 19, and let\({{\bf{\hat \beta }}_{\bf{n}}}\)n denote the M.L.E. of 尾.

a. Use the delta method to determine the asymptotic distribution of\(\frac{{\bf{1}}}{{{{{\bf{\hat \beta }}}_{\bf{n}}}}}\).

b. Show that\(\frac{{\bf{1}}}{{{{{\bf{\hat \beta }}}_{\bf{n}}}}}{\bf{ = }}{{\bf{\bar X}}_{\bf{n}}}\), and use the central limit theorem to determine the asymptotic distribution of\(\frac{{\bf{1}}}{{{{{\bf{\hat \beta }}}_{\bf{n}}}}}\).

Suppose that the five random variables \({{\bf{X}}_{\bf{1}}}{\bf{, \ldots ,}}{{\bf{X}}_{\bf{5}}}\) are i.i.d. and that each has the standard normal distribution. Determine a constantcsuch that the random variable

\(\frac{{{\bf{c}}\left( {{{\bf{X}}_{\bf{1}}}{\bf{ + }}{{\bf{X}}_{\bf{2}}}} \right)}}{{{{\left( {{\bf{X}}_{\bf{3}}^{\bf{2}}{\bf{ + X}}_{\bf{4}}^{\bf{2}}{\bf{ + X}}_{\bf{5}}^{\bf{2}}} \right)}^{\frac{{\bf{1}}}{{\bf{2}}}}}}}\)

will have atdistribution.

Suppose that\({X_1}...{X_n}\)form a random sample from the normal distribution with mean 0 and unknown standard deviation\(\sigma > 0\). Find the lower bound specified by the information inequality for the variance of any unbiased estimator of\(\log \sigma \).

Suppose that \({{\bf{X}}_{\bf{1}}}{\bf{, }}{\bf{. }}{\bf{. }}{\bf{. , }}{{\bf{X}}_{\bf{n}}}\) form a random sample from the normal distribution with unknown mean \({\bf{\mu }}\,\,{\bf{and}}\,\,{\bf{\tau }}\), and also that the joint prior distribution of \({\bf{\mu }}\,\,{\bf{and}}\,\,{\bf{\tau }}\) is the normal-gamma distribution satisfying the following conditions: \({\bf{E}}\left( {\bf{\mu }} \right){\bf{ = 0}}\,\,\,\,{\bf{,E}}\left( {\bf{\tau }} \right){\bf{ = 2,E}}\left( {{{\bf{\tau }}^{\bf{2}}}} \right){\bf{ = 5}}\,\,\,{\bf{and}}\,\,{\bf{Pr}}\left( {\left| {\bf{\mu }} \right|{\bf{ < 1}}{\bf{.412}}} \right){\bf{ = 0}}{\bf{.5}}\)Determine the prior hyperparameters \({{\bf{\mu }}_{\bf{0}}}{\bf{,}}{{\bf{\lambda }}_{\bf{0}}}{\bf{,}}{{\bf{\alpha }}_{\bf{0}}}{\bf{,}}{{\bf{\beta }}_{\bf{0}}}\)
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

Read Explanation

Calculus

Read Explanation

Logic and Functions

Read Explanation

Probability and Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.