   # In Exercises 10–12, use the given probability density function to find each probability. f ( x ) = 4 ( x − x 3 ) , [ 0 , 1 ] (a) P ( 0 &lt; x &lt; 0.5 ) (b) P ( 0.25 ≤ x &lt; 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 11TYS
Textbook Problem
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## In Exercises 10–12, use the given probability density function to find each probability. f ( x ) = 4 ( x − x 3 ) ,   [ 0 ,   1 ] (a) P ( 0 < x < 0.5 ) (b) P ( 0.25   ≤ x < 1 )

(a)

To determine

To calculate: The probability P(0<x<0.5) of the probability density function f(x)=4(xx3) over the interval [0,1].

### Explanation of Solution

Given Information:

The probability density function f(x)=4(xx3) over the interval [0,1].

Formula used:

If f is function of a continuous random variable x, then the probability that x lies in the interval [c,d] is,

P(cxd)=cdf(x) dx

Calculation:

Consider the provided probability density function f(x)=4(xx3).

Use the formula P(cxd)=cdf(x) dx to calculate the required probability P(0<x<0.5).

So,

P(0<x<0.5)=00.54(xx3) dx=400.5(xx3) dx

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

(b)

To determine

To calculate: The probability P(0.25x<1) of the probability density function f(x)=4(xx3) over the interval [0,1].

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