Explanation Given information: A function is given by p ( x ) = 1 π 1 − x 2 on the interval ( − 1 , 1 ) . Formula used: For a probability density function p ( x ) on ( − ∞ , ∞ ) , the mean μ of the random variable is given by μ = ∫ − ∞ ∞ x p ( x ) d x For a probability density function p ( x ) on ( − ∞ , ∞ ) , the standard deviation σ of the random variable is given by σ 2 = ∫ − ∞ ∞ ( x − μ ) 2 p ( x ) d x The integral below can be calculated as. ∫ x 1 − x 2 d x = − 1 − x 2 + C The integral below can be calculated as. ∫ x 2 1 − x 2 d x = 1 2 ( sin − 1 x − 1 2 sin ( 2 sin − 1 x ) ) + C Calculation: Let us consider the given probability density function. p ( x ) = 1 π 1 − x 2 Now calculate the mean of the probability density function on ( − 1 , 1 ) as shown below. μ = ∫ − 1 1 x p ( x ) d x = ∫ − 1 1 x ( 1 π 1 − x 2 ) d x = 1 π ∫ − 1 1 x 1 − x 2 d x = 1 π [ − 1 − x 2 ] − 1 1 = 1 π [ − 0 − 0 ] = 0 Mean of the probability density function is 0

Loose-leaf Version For Calculus: Early Transcendentals

4th Edition

ISBN: 9781319055912

Author: Jon Rogawski, Colin Adams, Robert Franzosa

Publisher: W. H. Freeman

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Chapter 8.1, Problem 28E

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The mean and standard deviation of the given probability density function p(x) on (−1,1).

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Chapter 8.1, Problem 27E

Chapter 8.1, Problem 29E

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Chapter 8 Solutions

Loose-leaf Version For Calculus: Early Transcendentals

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The mean and standard deviation of the given probability density function p ( x ) on ( − 1 , 1 ) .

Solution Summary: The author explains how to calculate the mean and standard deviation of the given probability density function p(x).

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To calculate:

The mean and standard deviation of the given probability density function p(x) on (−1,1).

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Chapter 8 Solutions

The mean and standard deviation of the given probability density function p ( x ) on ( − 1 , 1 ) .

Solution Summary: The author explains how to calculate the mean and standard deviation of the given probability density function p(x).

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To calculate: The mean and standard deviation of the given probability density function p(x) on (−1,1).

Want to see the full answer?

Chapter 8 Solutions

To calculate:

The mean and standard deviation of the given probability density function p(x) on (−1,1).