/*! 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} Q:5.3 For some constant c, the random ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Q:5.3 (page 217)

For some constant c, the random variable X has the probability density function:

f(x)=cx4鈥呪赌呪赌呪赌�0<x<20鈥呪赌呪赌呪赌�otherwise

Find

  1. E[X]and
  2. Var(X)

Short Answer

Expert verified

a. The value of E[X]=53

b. Var(X)=563

Step by step solution

01

Step:1 Given Information (Part a)

The random variable X has the probability density function

f(x)=cx4鈥呪赌呪赌呪赌�0<x<20鈥呪赌呪赌呪赌�otherwise

We have to findE[X]

02

Definition (part a)

The Probability Density Function (PDF) is a probability function that represents the density of a continuous random variable that falls inside a given range of values.

03

Step:3 Explanation (Part a)

Let's start by determining the c constant. There must be one,

1=鈭�鈩�f(x)dx=鈭�02cx4dx=cx5502=c路325

which implies that c=532

(a) The mean value is

E(X)=鈭�鈩�xf(x)dx=鈭�02x路532x4dx=532鈭�02x5dx=532x6602=532脳646=53

04

Explanation (Part b)

According to the information, We observe that:

EX2=鈭�02鈥�x2鈰�5x432dx=532x7702

53227鈭�0=532脳7脳128=207

Therefore Var(X):

localid="1650014593098" Var(X)=Ex2鈭�(E(x))2

localid="1649597273687" 鈬�Var(X)=207鈭�532=563

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

With 桅(x)being the probability that a normal random variable with mean 0and variance 1is less than x, which of the following are true:

(a)桅(-x)=桅(x)

(b)桅(x)+桅(-x)=1

(c)桅(-x)=1/桅(x)

Show that Zis a standard normal random variable; then, for,x>0

  1. P{Z>x}=P{Z<鈭�x}
  2. P{|Z|>x}=2P{Z>x}
  3. P{|Z|<x}=2P{Z<x}鈭�1

If Yis uniformly distributed over (0,5),what is the probability that the roots of the equation 4x2+4xY+Y+2=0are both real?

If xis uniformly distributed over (a,b), what random variable, having a linear relation with x, is uniformly distributed over (0,1)?

There are two types of batteries in a bin. When in use, type i batteries last (in hours) an exponentially distributed time with rate 位i,i=1,2. A battery that is randomly chosen from the bin will be a type i battery with probability pi, pi,鈭�i=12鈥�pi=1. If a randomly chosen battery is still operating after t hours of use, what is the probability that it will still be operating after an additional shours?
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

Read Explanation

Discrete Mathematics

Read Explanation

Calculus

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.