Explanation Given information: The density function of a random variable X is f ( x ) = 1 18 x , 0 ≤ x ≤ 6 . Formula used: 1) E ( X ) = ∫ A B x f ( x ) d x . 2) Var ( X ) = ∫ A B x 2 f ( x ) d x − [ E ( X ) ] 2 . Calculation: A random variable X with the density function f ( x ) = 1 18 x , 0 ≤ x ≤ 6 1) E ( X ) = ∫ A B x f ( x ) d x . Plug limits A = 0 , B = 6 , and f ( x ) = 1 18 x in the expected value formula, E ( X ) = ∫ 0 6 x 1 18 x d x = ∫ 0 6 1 18 x 2 d x Using the power rule of integration, ⇒ E ( X ) = 1 18 ⋅ ( x 3 3 ) | 0 6 = x 3 54 | 0 6 = 6 3 54 − 0 3 54 = 216 54 = 4 2) Var ( X ) = ∫ A B x 2 f ( x ) d x − [ E ( X ) ] 2

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Chapter 12.3, Problem 1E

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To calculate: The expected value E(X) and variance Var(X) with the density function f(x)=118x, 0≤x≤6.

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Chapter 12.3, Problem 2CYU

Chapter 12.3, Problem 2E

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The expected value E ( X ) and variance Var ( X ) with the density function f ( x ) = 1 18 x , 0 ≤ x ≤ 6 .

Solution Summary: The author explains how to calculate the expected value and variance of a random variable X with the density function.

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To calculate: The expected value E(X) and variance Var(X) with the density function f(x)=118x, 0≤x≤6.

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Chapter 12 Solutions