/*! 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} Q1 E Assume that聽X1, . . . , Xn聽fro... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Q1 E (page 479)

Assume thatX1, . . . , Xnfrom a random sample from the normal distribution with mean渭and variance \({\sigma ^2}\). Show that \({\hat \sigma ^2}\)has the gamma distribution with parameters \(\frac{{\left( {n - 1} \right)}}{2}\)and\(\frac{n}{{\left( {2{\sigma ^2}} \right)}}\).

Short Answer

Expert verified

\({\hat \sigma ^2}\) follows Gamma distribution with parameters \(\frac{{n - 1}}{2}\) and \(\frac{n}{{2{\sigma ^2}}}\).

Step by step solution

01

Given information

\({X_1},{X_2},...,{X_n}\) are normal random variables.

02

Determine the distribution of \({\hat \sigma ^2}\) 

Let \(U = \frac{{n{{\hat \sigma }^2}}}{{{\sigma ^2}}}\) follows \({\chi ^2}\)a distribution with degrees of freedom \(n - 1\).i.e,

\(U\)follows Gamma distribution with parameters \(\frac{{n - 1}}{2}\) and \(\frac{1}{2}\).

Let, \(c = \frac{{{\sigma ^2}}}{n}\)

Then,

\(\begin{align}U &= \frac{{n{{\hat \sigma }^2}}}{{{\sigma ^2}}}\\ &= \frac{1}{c}{{\hat \sigma }^2}\\ \Rightarrow cU &= {{\hat \sigma }^2}\end{align}\)

Now,\(cU\) follows Gamma distribution with parameters \(\frac{{n - 1}}{2}\) and \(\frac{n}{{2{\sigma ^2}}}\).

Hence,\({\hat \sigma ^2}\) follows Gamma distribution with parameters \(\frac{{n - 1}}{2}\) and \(\frac{n}{{2{\sigma ^2}}}\).

Hence, proved.

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

Complete the proof of Theorem 8.5.3 by dealing with the case in which r(v, x) is strictly decreasing in v for each x.

Suppose that two random variables\({\bf{\mu }}\,\,{\bf{and}}\,\,{\bf{\tau }}\)have the joint normal-gamma distribution with hyperparameters\({{\bf{\mu }}_{\bf{0}}}{\bf{ = 4,}}{{\bf{\lambda }}_{\bf{0}}}{\bf{ = 0}}{\bf{.5,}}{{\bf{\alpha }}_{\bf{0}}}{\bf{ = 1,}}{{\bf{\beta }}_{\bf{0}}}{\bf{ = 8}}\)Find the values of (a)\({\bf{Pr}}\left( {{\bf{\mu > 0}}} \right)\)and (b)\({\bf{Pr}}\left( {{\bf{0}}{\bf{.736 < \mu < 15}}{\bf{.680}}} \right)\).

Suppose that\({{\bf{X}}_{\bf{1}}}{\bf{,}}{{\bf{X}}_{\bf{2}}}{\bf{,}}...{\bf{,}}{{\bf{X}}_{\bf{n}}}\)form a random sample from the uniform distribution on the interval\(\left( {{\bf{0,1}}} \right)\), and let\({\bf{W}}\)denote the range of the sample, as defined in Example 3.9.7. Also, let\({{\bf{g}}_{\bf{n}}}\left( {\bf{x}} \right)\)denote the p.d.f of the random

variable\({\bf{2n}}\left( {{\bf{1 - W}}} \right)\), and let\({\bf{g}}\left( {\bf{x}} \right)\)denote the p.d.f of the\({\chi ^{\bf{2}}}\)distribution with four degrees of freedom. Show that

\(\mathop {{\bf{lim}}}\limits_{{\bf{n}} \to \infty } {{\bf{g}}_{\bf{n}}}\left( {\bf{x}} \right){\bf{ = g}}\left( {\bf{x}} \right)\) for\({\bf{x > 0}}\).

Prove that the distribution of\({\hat \sigma _0}^2\)in Examples 8.2.1and 8.2.2 is the gamma distribution with parameters\(\frac{n}{2}\)and\(\frac{n}{{2{\sigma ^2}}}\).

Consider the analysis performed in Example 8.6.2. This time, use the usual improper before computing the parameters' posterior distribution.
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Logic and Functions

Read Explanation

Geometry

Read Explanation

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Decision Maths

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.