Explanation Given: The function f ( x ) = x 8 − 1 4 is a probability density function on the interval [ 2 , 6 ] . 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 ) = x 8 − 1 4 on the interval [ 2 , 6 ] . Use the above mentioned definition and find expectation. E ( X ) = μ = ∫ 2 6 x f ( x ) d x = ∫ 2 6 x ( x 8 − 1 4 ) d x = ∫ 2 6 ( x 2 8 − x 4 ) d x On further simplification the above expression becomes, E ( X ) = [ x 3 24 − x 2 8 ] 2 6 = 1 24 [ 6 3 − 2 3 ] − 1 8 [ 6 2 − 2 2 ] = 1 24 [ 216 − 8 ] − 1 8 [ 36 − 4 ] = 1 24 × 208 − 1 8 × 32 On further simplification the above expression becomes, E ( X ) = 26 3 − 4 = 14 3 ≈ 4

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Chapter 11.2, Problem 3E

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To find: The expected value, variance and standard deviation for the function f(x)=x8−14 .

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Chapter 11.2, Problem 2E

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The expected value, variance and standard deviation for the function f ( x ) = x 8 − 1 4 .

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To find: The expected value, variance and standard deviation for the function f(x)=x8−14 .

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