Explanation Given information: A function is given by p ( x ) = 5 2 x 7 / 2 on the interval [ 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 power rule of indefinite integrals. ∫ x n d x = x n n + 1 + C Calculation: Let us consider the given probability density function. p ( x ) = 5 2 x 7 / 2 Now calculate the mean of the probability density function on [ 1 , ∞ ) as shown below. μ = ∫ 1 ∞ x p ( x ) d x = ∫ 1 ∞ x ( 5 2 x 7 / 2 ) d x = 5 2 ∫ 1 ∞ x − 5 / 2 d x = 5 2 lim A → ∞ ∫ 1 A x − 5 / 2 d x = 5 2 lim A → ∞ [ 2 x − 3 / 2 − 3 ] 1 A = − 5 2 × 2 3 lim A → ∞ [ x − 3 / 2 ] 1 A = − 5 3 lim A → ∞ [ 1 A 3 / 2 − 1 ] = − 5 3 [ 0 − 1 ] = 5 3 Mean of the probability density function is 5 3 . Now calculate the standard deviation of the probability density function on [ 1 , ∞ ) as shown below

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ISBN: 9781319050740

Author: Jon Rogawski, Colin Adams, Robert Franzosa

Publisher: W. H. Freeman

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Chapter 8.1, Problem 27E

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

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Chapter 8.1, Problem 26E

Chapter 8.1, Problem 28E

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Chapter 8 Solutions

Calculus: Early Transcendentals

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

Solution Summary: The author calculates the mean and standard deviation of a given probability density function on p(x).

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To calculate:

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

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Chapter 8 Solutions

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

Solution Summary: The author calculates the mean and standard deviation of a given probability density function on p(x).

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