   # Finding Expected Value, Variance, and Standard Deviation In Exercises 39–44, use the given probability density function over the indicated interval to find the (a) expected value, (b) variance, and (c) standard deviation of the random variable. (d) Then sketch the graph of the probability density function and locate the mean on the graph. f ( x ) = 8 − x 32 , [ 0 , 8 ] ### 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 42RE
Textbook Problem
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## Finding Expected Value, Variance, and Standard Deviation In Exercises 39–44, use the given probability density function over the indicated interval to find the (a) expected value, (b) variance, and (c) standard deviation of the random variable. (d) Then sketch the graph of the probability density function and locate the mean on the graph. f ( x ) = 8 − x 32 , [ 0 , 8 ]

(a)

To determine

To calculate: The expected value of probability density function f(x)=8x32 over the interval [0,8].

### Explanation of Solution

Given Information:

The probability density function is, f(x)=8x32 over the interval [0,8].

Formula used:

The expected value E(x) of a probability density function f for a continuous random variable x over the interval [a,b] is,

μ=E(x)=abxf(x) dx

The simple power rule of integration,

xndx=xn+1n+1+C

The Fundamental theorem of calculus for definite integral of a function f on a closed interval [a,b] is,

abf(x)dx=F(b)F(a).

Calculation:

Consider the provided probability density function, f(x)=8x32.

Use the definition of expected value for the provided probability density function,

E(x)=abxf(x) dx

Substitute 0 for a, 8 for b and 8x32 for f(x)

(b)

To determine

To calculate: The variance of probability density function f(x)=8x32 over the interval [0,8].

(c)

To determine

To calculate: The standard deviation of probability density function f(x)=8x32 over the interval [0,8].

(d)

To determine

To graph: The probability density function f(x)=8x32 over the interval [0,8].

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