Explanation Given information: A function is given by p ( x ) = 1 3 e − x / 3 on the interval [ 0 , ∞ ) . Formula used: For an exponential probability density function of the form p ( t ) = 1 r e − t r on [ 0 , ∞ ) , the mean r such that r > 0 . 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 following integration by parts can be calculated as shown below. ∫ x 2 e − x / a d x = − a x 2 e − x / a + 2 a ( − a x e − x / a − a 2 e − x / a ) + C ∫ x e − x / a d x = ( − a x e − x / a − a 2 e − x / a ) + C ∫ e − x / a d x = ( − a e − x / a ) + C Calculation: Let us consider the given probability density function. p ( x ) = 1 3 e − x / 3 The function is exponential. Let us compare the function with the form p ( x ) = 1 r e − x r . Now we know that mean of exponential; density function of the above form will be r . So, here r = 3 Mean of the probability density function is 3 . Now calculate the standard deviation of the probability density function on [ 0 , ∞ ) as shown below

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 29E

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The mean and standard deviation of the given probability density function p(x) on [0,∞).

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Chapter 8.1, Problem 28E

Chapter 8.1, Problem 30E

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Chapter 8 Solutions

Loose-leaf Version For Calculus: Early Transcendentals

Ch. 8.1 - Prob. 1PQ Ch. 8.1 - Prob. 2PQ Ch. 8.1 - Prob. 3PQ Ch. 8.1 - Prob. 1E Ch. 8.1 - Prob. 2E Ch. 8.1 - Prob. 3E Ch. 8.1 - Prob. 4E Ch. 8.1 - Prob. 5E Ch. 8.1 - Prob. 6E Ch. 8.1 - Prob. 7E

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The mean and standard deviation of the given probability density function p ( x ) on [ 0 , ∞ ) .

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 [0,∞).

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Chapter 8 Solutions

The mean and standard deviation of the given probability density function p ( x ) on [ 0 , ∞ ) .

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 [0,∞).

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Chapter 8 Solutions

To calculate:

The mean and standard deviation of the given probability density function p(x) on [0,∞).