Logout Open In App
Open In App
Warning: foreach() argument must be of type array|object, bool given in /var/www/html/web/app/themes/studypress-core-theme/template-parts/header/mobile-offcanvas.php on line 20

Chapter 8: Q4E (page 527)

Suppose a random variable has a normal distribution with a mean of 0 and an unknown standard deviation σ> 0. Find the Fisher information I (σ) in X.

Short Answer

Expert verified

The fisher information is 2n/ σ2

Step by step solution

01

Given the information

It is given that X is a random variable that follows Normal distribution with a known mean of 0 and unknown standard deviation >0. Therefore X1,….., Xn are iid Normal (µ=0, σ>0) .

02

Define the pdf


03

Define fisher information


04

Calculating fisher information for normal distribution

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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

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

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\({\bf{x > 0}}\).

Question:Suppose that a random variable X has a normal distribution for which the mean μ is unknown (−∞ <μ< ∞) and the variance σ2 is known. Let\({\bf{f}}\left( {{\bf{x}}\left| {\bf{\mu }} \right.} \right)\)denote the p.d.f. of X, and let\({\bf{f'}}\left( {{\bf{x}}\left| {\bf{\mu }} \right.} \right)\)and\({\bf{f''}}\left( {{\bf{x}}\left| {\bf{\mu }} \right.} \right)\)denote the first and second partial derivatives with respect to μ. Show that

\(\int_{{\bf{ - }}\infty }^\infty {{\bf{f'}}\left( {{\bf{x}}\left| {\bf{\mu }} \right.} \right)} {\bf{dx = 0}}\,\,{\bf{and}}\,\,\int_{{\bf{ - }}\infty }^\infty {{\bf{f''}}\left( {{\bf{x}}\left| {\bf{\mu }} \right.} \right)} {\bf{dx = 0}}{\bf{.}}\).

Sketch the p.d.f. of the\({{\bf{\chi }}^{\bf{2}}}\)distribution withmdegrees of freedom for each of the following values ofm. Locate the mean, the median, and the mode on each sketch. (a)m=1;(b)m=2; (c)m=3; (d)m=4.

Suppose thatXhas thetdistribution withmdegrees of freedom(m >2). Show that Var(X)=m/(m−2).

Hint:To evaluate\({\bf{E}}\left( {{{\bf{X}}^{\bf{2}}}} \right)\), restrict the integral to the positive half of the real line and change the variable fromxto

\({\bf{y = }}\frac{{\frac{{{{\bf{x}}^{\bf{2}}}}}{{\bf{m}}}}}{{{\bf{1 + }}\frac{{{{\bf{x}}^{\bf{2}}}}}{{\bf{m}}}}}\)

Compare the integral with the p.d.f. of a beta distribution. Alternatively, use Exercise 21 in Sec. 5.7.

Consider again the situation described in Example 8.2.3. How small wouldσ2 need to be in order for Pr(Y≤0.09)≥0.9?
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Applied Mathematics

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.

Sign-up for free