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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 Technology In Exercises 17-22, use a symbolic integration utility to find the mean, variance, and standard deviation of the probability density function. Then find the percent of the distribution that lies within the given standard deviations of the mean. See Example 3.

Function StandardDeviations

f ( x ) = 1 2 π e x 2 / 2 ,  - < x <            1 .5

To determine

To calculate: The mean, variance and standard deviation of the probability density function f(x)=12πex22;<x<. And also find the percentage of the distribution that lies within the given standard deviation, 1.5 of the mean.

Explanation

Given Information:

The probability distribution function is, f(x)=12πex22;<x<. The standard deviation is equal to 1.5.

Formula used:

The definition of mean over the interval [a,b],

μ=E(x)=abxf(x)dx

Where, f is a probability density function and x is a continuous random variable.

The definition of variance is,

V(x)=ab(xμ)2f(x)dx

The definition of standard deviation is

σ=V(x)

Calculation:

Consider the probability distribution function,

f(x)=12πex22;<x<

The mean of the function is defined as,

μ=abxf(x)dx=x12πex22dx

By the use of symbolic integration utility to find the value of mean.

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