Explanation Definition used: If f ( x ) is the probability density function for continuous random variable X , then The Mean is, μ = E ( X ) = ∫ − ∞ ∞ x f ( x ) d x . The variance is, V ( X ) = ∫ − ∞ ∞ ( x − μ ) 2 f ( x ) d x . The standard deviation is, σ = V ( X ) . Calculation: The given function is f ( x ) = { 4 − 2 x if 1 ≤ x ≤ 2 0 otherwise . Obtain the mean of the function by using above definition. μ = ∫ − ∞ ∞ x f ( x ) d x = ∫ 1 2 ( 4 x − 2 x 2 ) d x = 4 [ x 2 2 ] 1 2 − 2 [ x 3 3 ] 1 2 = 4 [ 4 2 − 1 2 ] − 2 [ 8 3 − 1 3 ] On further simplification, μ = 12 2 − 14 3 = 36 − 28 6 = 8 6 = 4 3 Thus, the mean is 4 3 . Substitute μ = 4 3 in V ( X ) = ∫ − ∞ ∞ ( x − μ ) 2 f ( x ) d x and obtain the variance of the function as follows, V ( X ) = ∫ − ∞ ∞ ( x − μ ) 2 f ( x )

Calculus for Business, Economics, Life Sciences, and Social Sciences (14th Edition)

14th Edition

ISBN: 9780134668574

Author: Raymond A. Barnett, Michael R. Ziegler, Karl E. Byleen, Christopher J. Stocker

Publisher: PEARSON

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Chapter 11.3, Problem 5E

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To find: The mean, variance, and standard deviation for the function f(x)={4−2x if 1≤x≤20 otherwise.

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Chapter 11.3, Problem 4E

Chapter 11.3, Problem 6E

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Chapter 11 Solutions

Calculus for Business, Economics, Life Sciences, and Social Sciences (14th Edition)

Ch. 11.1 - Evaluate the following, if it converges: 3dx(x1)2.Ch. 11.1 - Prob. 2MP Ch. 11.1 - Prob. 3MP Ch. 11.1 - Prob. 4MP Ch. 11.1 - Prob. 5MP Ch. 11.1 - Prob. 6MP Ch. 11.1 - Prob. 1ED Ch. 11.1 - Prob. 2ED Ch. 11.1 - Prob. 1E Ch. 11.1 - Prob. 2E

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The mean, variance, and standard deviation for the function f ( x ) = { 4 − 2 x if 1 ≤ x ≤ 2 0 otherwise .

Solution Summary: The author calculates the mean, variance, and standard deviation for the function f(x).

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To find: The mean, variance, and standard deviation for the function f(x)={4−2x if 1≤x≤20 otherwise.

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Chapter 11 Solutions