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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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Finding Expected Value, Variance, and Standard Deviation In Exercises 1-12, use the given probability density function over the indicated interval to find the (a) expected value,

(b) variance using the alternative formula, 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. See Examples 1, 2, and 5.

f ( x ) = 6 x ( 1 x ) , [ 0 , 1 ]

(a)

To determine

To calculate: The expected value for the probability density function, f(x)=6x(1x) on the interval [0,1].

Explanation

Given Information:

The provided probability density function is f(x)=6x(1x) on the interval [0,1].

Formula used:

The expected value for the continuous random variable, x with density function f(x) within the interval [a,b] is computed as:

μ=E(x)=abxf(x)dx

Calculation:

Consider the provided probability density function, f(x)=6x(1x) on the interval [0,1].

So, a=0 and b=1.

Substitute the values a=0, b=1 and f(x)=6x(1x) in the formula μ=abxf(x)dx

(b)

To determine

To calculate: The variance for the probability density function, f(x)=6x(1x) on the interval [0,1].

(c)

To determine

To calculate: The standard deviation for the probability density function, f(x)=6x(1x) on the interval [0,1].

(d)

To determine

To graph: The probability density function, f(x)=6x(1x) on the interval [0,1] and mark the mean value.

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