   # In Exercises 13–15, use the given probability density function over the indicated interval to find the expected value, variance, and standard deviation of the random variable. f ( x ) = 1 14 , [ 0 , 14 ] ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
Publisher: Cengage Learning
ISBN: 9781305860919

#### Solutions

Chapter
Section ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
Publisher: Cengage Learning
ISBN: 9781305860919
Chapter 9, Problem 13TYS
Textbook Problem
1 views

## In Exercises 13–15, use the given probability density function over the indicated interval to find the expected value, variance, and standard deviation of the random variable. f ( x ) = 1 14 ,   [ 0 , 14 ]

To determine

To calculate: The expected value, variance and standard deviation of probability density function f(x)=114 over the interval [0,14].

### Explanation of Solution

Given Information:

The probability density function is defined as f(x)=114 over the interval [0,14].

Formula used:

For any probability density function f of a continuous random variable x over the interval [a,b], the expected value of x is defined as,

μ=E(x)=abxf(x) dx

For any probability density function f of a continuous random variable x over the interval [a,b], the variance of x is defined as,

V(x)=ab(xμ)2f(x) dx

Here μ is the mean or the expected value of x.

For any probability density function f of a continuous random variable x over the interval [a,b], the standard deviation of x is defined as,

σ=V(x)

Calculation:

Consider the provided probability density function.

f(x)=114 over the interval [0,14].

Use the formula E(x)=abxf(x) dx for the provided probability density function to calculate the expected value.

So,

E(x)=014x114 dx=114014x dx

Use the formula xndx=xn+1n+1 and integrate.

114014x dx=114(x22)014=128[x2]014

Use the Fundamental theorem abf(x) dx=F(b)F(a) and apply the limits.

128[x2]014=128=128(196)=19628=7

Thus, the expected value of the provided probability density function is 7.

Now calculate the variance.

The mean of the probability density function is obtained as 7.

Now use the formula V(x)=ab(xμ)2f(x) dx for the provided probability density function to calculate the variance

### Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

#### The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

Find more solutions based on key concepts
Solve the equations in Exercises 126. x44x2=4

Finite Mathematics and Applied Calculus (MindTap Course List)

f(x)1xx24

Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach

In problems 13-16, Use notation to indicate which set is a subset of the other. 15.

Mathematical Applications for the Management, Life, and Social Sciences

Subtract: (+20)(30)

Elementary Technical Mathematics

Given: mRST=39 mTSV=23 Find: mRSV Exercises 1624

Elementary Geometry for College Students

True or False: Predator-prey population models are the solutions to two differential equations.

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th

For

Study Guide for Stewart's Multivariable Calculus, 8th 