   # 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 ) = 3 x − 3 2 x 2 , [ 0 , 1 ] ### 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 14TYS
Textbook Problem
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## 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 ) = 3 x − 3 2 x 2 , [ 0 , 1 ]

To determine

To calculate: The expected value, variance and standard deviation of probability density function f(x)=3x32x2 over the interval [0,1].

### Explanation of Solution

Given Information:

The probability density function is defined as f(x)=3x32x2 over the interval [0,1].

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)=3x32x2 over the interval [0,1].

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

So,

E(x)=01x(3x32x2) dx=01(3x232x3) dx

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

01(3x232x3) dx=(3x3332(x44))01=[x33x48]01

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

[x33x48]01=[(133(14)8)0]=138=58

Thus, the expected value of the provided probability density function is 58.

Now calculate the variance.

The mean of the probability density function is obtained as 58.

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

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