   Chapter 9.2, Problem 16E ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

#### Solutions

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

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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# Making a Probability Density Function In Exercises 13–18, find the constant k such that the function f is a probability density function over the given interval. f ( x ) = k x ( 1 − x ) ,     [ 0 , 1 ]

To determine

To calculate: The value of constant ‘k’ for which the function, f(x)=kx(1x) is a probability density function over the interval [0,1].

Explanation

Given Information:

The function f(x)=kx(1x) is a probability density function over the interval [0,1].

Formula used:

Consider a function f of a continuous random variables x whose set of values is the interval [a,b], A function is a probability density function when it is non-negative and continuous on the interval [a,b] and when,

abf(x)dx=1

Calculation:

Consider the function,

f(x)=kx(1x)

As the function is a probability density function over the interval [0,1].

Therefore, f(x) is continuous and non-negative over the interval [0,1].

Also 01f(x)dx=1 as,

01f(x)dx=101kx(1x) dx=1k01(xxx)dx=1k01(x12x32)dx=1

Further simplify,

k[01(x12)dx01(x32)dx<

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