Explanation Given information: X be a continuous random variable with values between A = 1 and B = ∞ with density function f ( x ) = 4 x − 5 . Formula used: Let X be a continuous random variable whose possible values lie between A and B , and f ( x ) be the probability density function for X . Then, the expected value (average) of X is the number E ( X ) defined by, E ( X ) = ∫ A B x f ( x ) d x . And the variance of X is defined by, Var ( X ) = ∫ A B [ x − E ( X ) ] 2 f ( x ) d x . Calculation: Let X be the continuous random variable with values between A = 1 and B = ∞ with density function, f ( x ) = 4 x − 5 . By using the formula of expected value, E ( X ) = ∫ 1 ∞ x f ( x ) d x = ∫ 1 ∞ x 4 x − 5 d x . = ∫ 1 ∞ 4 x − 4 d x . The upper limit of integration is ∞ that is indefinite form of integration, so take limit as b → ∞ of the integration. E ( X ) = lim b → ∞ ∫ 1 b 4 x − 4 d x Apply the power rule of integration, ⇒ E ( X ) = lim b → ∞ [ 4 x − 3 − 3 ] | 1 b = lim b → ∞ [ − 4 3 x − 3 ] 1 b = lim b → ∞ [ − 4 3 b − 3 + 4 3 1 − 3 ] = lim b → ∞ [ − 4 3 b − 3 + 4 3 ] Apply limit as b → ∞ = [ − 4 3 ∞ − 3 + 4 3 ] = 0 + 4 3 ⇒ E ( X ) = 4 3 . By using the formula of variance, Var ( X ) = ∫ 1 ∞ [ x − E ( X ) ] 2 f ( x ) d x = ∫ 1 ∞ [ x − 4 3 ] 2 ⋅ 4 x − 5 d x The upper limit of integration is ∞ that is indefinite form of integration, so take limit as b → ∞ of the integration

12th Edition

ISBN: 9780137590810

Author: Larry J. Goldstein, David C. Lay, David I. Schneider, Nakhle H. Asmar, William Edward Tavernetti

Publisher: PEARSON

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Question

Chapter 12.3, Problem 17E

To determine

To calculate: E(X) and Var(X) of the density function f(x) = 4x−5

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Chapter 12.3, Problem 16E

Chapter 12.3, Problem 18E

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E ( X ) and Var ( X ) of the density function f ( x ) = 4 x − 5

Solution Summary: The author calculates X as a continuous random variable with values between A and B with density function.

To calculate: E(X) and Var(X) of the density function f(x) = 4x−5

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Chapter 12 Solutions