To determine To find: The expected value, variance and standard deviation for the function f ( x ) = 4 x − 5 . Explanation Given: The function f ( x ) = 4 x − 5 is a probability density function on the interval [ 1 , ∞ ) . Definition used: “If X is a continuous random variable with probability function f on [ a , b ] , then the expected value of X is E ( X ) = μ = ∫ a b x f ( x ) d x The variance of X is V a r ( X ) = ∫ a b x 2 f ( x ) d x − ( μ ) 2 . And the standard deviation of X is σ ≈ V a r ( X ) ”. Calculation: Consider the given function f ( x ) = 4 x − 5 on the interval [ 1 , ∞ ) . Use the above mentioned definition and find expectation. E ( X ) = μ = ∫ 1 ∞ x f ( x ) d x = 4 ∫ 1 ∞ x × x − 5 d x = 4 ∫ 1 ∞ x − 4 d x On further simplification the above expression becomes, E ( X ) = 4 lim a → ∞ ∫ 1 a x − 4 d x = 4 lim a → ∞ [ x − 4 + 1 − 4 + 1 ] 1 a = − 4 3 lim a → ∞ [ x − 3 ] 1 a = − 4 3 [ a − 3 − 1 − 3 ] On further simplification the above expression becomes, E ( X ) = − 4 3 ( 0 − 1 ) ( ∵ ∞ − 3 = 0 ) = 4 3 ≈ 1

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Chapter 11.2, Problem 7E

To determine

To find: The expected value, variance and standard deviation for the function f(x)=4x−5 .

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Chapter 11.2, Problem 6E

Chapter 11.2, Problem 8E

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