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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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Using Two Methods In Exercises 13-16, find the variance of the probability density function using two methods.

f ( x ) = 1 4 x + 1 4 ,       [ 0 , 2 ]

To determine

To calculate: The variance of the probability density function f(x)=x4+14 on the interval [0,2] by the use of two methods.

Explanation

Given Information:

The provided probability density function is f(x)=x4+14 in the interval [0,2].

Formula used:

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

V(x)=ab(xu)2f(x)dx

Where, μ is the mean of x.

The alternative formula used to find the variance is given as,

V(x)=abx2f(x)dxu2

The formula used for expected value or mean is given as,

μ=abxf(x)dx

Calculation:

Consider the probability density function, f(x)=x4+14, where x[0,2].

Now, first calculate the mean μ of x by the use of formula,

μ=abxf(x)dx

Substitute f(x)=x4+14 and a=0, b=2 in the above formula,

μ=02x[x4+14]dx=02(x24+x4)dx=[x34×3+x24×2]02=[x312+x28]02

Simplify the above equation,

μ=[x312+x28]02=(2312+228)(012+08)=812+48=76

Consider the formula used for finding the variance,

V(x)=ab(xu)2f(x)dx

Substitute f(x)=x4+14, mean μ=76 and a=0, b=2 in the above equation,

V(x)=02(x76)2(x4+14

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