/*! 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} Q4.4-5E Suppose that X聽is a random vari... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Q4.4-5E (page 240)

Suppose that Xis a random variable with mean渭and variance\({\sigma ^2}\), and that the fourth moment ofXis finite. Show that \(E\left[ {{{\left( {X - \mu } \right)}^4}} \right] \ge {\sigma ^4}\).

Short Answer

Expert verified

\(E\left[ {{{\left( {X - \mu } \right)}^4}} \right] \ge {\sigma ^4}\)

Step by step solution

01

Given information

Let X be a random variable with mean \(\mu \) and variance\(\left( {{\sigma ^2}} \right)\).

The fourth moment of X is finite.

02

State and proof

Let, \(Y = {\left( {X - \mu } \right)^2}\)

Referring to the Exercise 4 for the information given below,

\(\begin{array}{c}Var\left( X \right) \ge 0\\E\left( {{X^2}} \right) - {\left[ {E\left( X \right)} \right]^2} \ge 0\\E\left( {{X^2}} \right) \ge {\left[ {E\left( X \right)} \right]^2}\end{array}\)

So,

\(\begin{array}{l}E\left( {{Y^2}} \right) \ge {\left[ {E\left( Y \right)} \right]^2}\\E\left[ {{{\left( {X - \mu } \right)}^4}} \right] \ge {\left[ {Var\left( X \right)} \right]^2}\\E\left[ {{{\left( {X - \mu } \right)}^4}} \right] \ge {\sigma ^4}\end{array}\)

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

Determine the value of a that a person would choose in Exercise 7 if his utility function was\({\bf{U}}\left( {\bf{x}} \right){\bf{ = x}}\)for\({\bf{x}} \ge {\bf{0}}\).

Determine which of the three gambles in Exercise 2 would be preferred by a person whose utility function has the form\({\bf{U}}\left( {\bf{x}} \right){\bf{ = ax + b}}\)where a and b are constants\(\left( {{\bf{a > 0}}} \right)\).

Suppose that a person's score X on a mathematics aptitude test is a number in the interval\(\left( {0,1} \right)\)and that his score Y on a music aptitude test is also a number in the interval\(\left( {0,1} \right)\)Suppose also that in the population of all college students in the United States, the scores X and Y are distributed in accordance with the following joint p.d.f:

\(f\left( {x,y} \right) = \left\{ \begin{align}\frac{2}{5}\left( {2x + 3y} \right)\;\;\;\;\;\;\;for\,0 \le x \le 1\,and0 \le x \le 1\\0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;otherwise\end{align} \right.\)

a. If a college student is selected randomly, what predicted value of his score on the music test has the smallest M.S.E.?

b. What predicted value of his score on the mathematics test has the smallest M.A.E.?

Suppose that a random variable X has a continuous distribution for which the pdf is as follows:

\(f\left( x \right) = \left\{ {\begin{aligned}{{}{}}{{e^{ - x}}}&{{\rm{for }}x > 0}\\0&{{\rm{otherwise}}}\end{aligned}} \right.\)

Determine all the medians of this distribution.

Suppose that on each play of a certain game a gambleris equally likely to win or to lose. Suppose that when hewins, his fortune is doubled and that when he loses, his fortune is cut in half. If he begins playing with a given

fortunec, what is the expected value of his fortune afternindependent plays of the game?
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Statistics

Read Explanation

Probability and Statistics

Read Explanation

Pure Maths

Read Explanation

Decision Maths

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.