Explanation Given: The function f ( x ) = 1 − 1 x is a probability density function on the interval [ 1 , 4 ] . 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 ) = 1 − 1 x on the interval [ 1 , 4 ] . Use the above mentioned definition and find expectation. E ( X ) = μ = ∫ 1 4 x f ( x ) d x = ∫ 1 4 x ( 1 − 1 x ) d x = ∫ 1 4 ( x − x 1 2 ) d x On further simplification the above expression becomes, E ( X ) = [ x 2 2 − x 1 2 + 1 1 2 + 1 ] 1 4 = 1 2 [ 4 2 − 1 2 ] − 2 3 [ 4 3 2 − 1 3 2 ] = 1 2 [ 16 − 1 ] − 2 3 [ ( 2 2 ) 3 2 − 1 ] = 1 2 × 15 − 2 3 [ 2 3 − 1 ] On further simplification the above expression becomes, E ( X ) = 15 2 − 2 3 × 7 = 15 2 − 14 3 = 17 6 ≈ 2

Calculus with Applications Books a la Carte Edition

11th Edition

ISBN: 9780133864564

Author: Margaret L. Lial; Nathan P. Ritchey; Raymond N. Greenwell

Publisher: Pearson Education

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Chapter 11.2, Problem 5E

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To find: The expected value, variance and standard deviation for the function f(x)=1−1x .

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Chapter 11.2, Problem 4E

Chapter 11.2, Problem 6E

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Chapter 11 Solutions

Calculus with Applications Books a la Carte Edition

Ch. 11.1 - YOUR TURN 1 Repeat Example 1(a) for the function...Ch. 11.1 - Prob. 2YT Ch. 11.1 - Prob. 3YT Ch. 11.1 - Prob. 4YT Ch. 11.1 - Prob. 1WE Ch. 11.1 - Prob. 2WE Ch. 11.1 - Prob. 3WE Ch. 11.1 - Decide whether the functions defined as follows...Ch. 11.1 - Decide whether the functions defined as follows...Ch. 11.1 - Decide whether the functions defined as follows...

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

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To find: The expected value, variance and standard deviation for the function f(x)=1−1x .

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Chapter 11 Solutions