a. Explanation Given info: Let X be the random variable with mean μ and standard deviation σ and g ( X ) be the function of X . The first order Tylor series approximation to g ( X ) in the neighborhood of μ is given by: g ( X ) ≈ g ( μ ) + g ′ ( μ ) . ( X − μ ) Where, g ( X ) is the linear function. Calculation: The expression for E [ g ( X ) ] is obtained as shown below: E [ g ( X ) ] = E [ g ( μ ) + g ′ ( μ ) . ( X − μ ) ] = E [ g ( μ ) ] + E [ g ′ ( μ ) . ( X − μ ) ] = E [ g ( μ ) ] + g ′ ( μ ) E [ X − μ ] Based on the rule of expected value, E ( X − μ ) = 0 E [ g ( X ) ] = E [ g ( μ ) ] + g ′ ( μ ) ( 0 ) = E [ g ( μ ) ] ≈ g ( μ ) [ Since, g ( μ ) is constant ] Thus, the expression for E [ g ( X ) ] is E [ g ( X ) ] ≈ g ( μ ) _ . The expression for V [ g ( X ) ] is obtained as shown below: V [ g ( X ) ] = V [ g ( μ ) + g ′ ( μ ) b. To determine Calculate the approximations to μ R and σ R .

Probability and Statistics for Engineering and the Sciences

9th Edition

ISBN: 9781305251809

Author: Jay L. Devore

Publisher: Cengage Learning

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Chapter 4, Problem 124SE

To determine

Obtain the expression for E[g(X)] and V[g(X)].

To determine

Calculate the approximations to μR and σR.

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Chapter 4, Problem 123SE

Chapter 4, Problem 125SE

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Obtain the expression for E [ g ( X ) ] and V [ g ( X ) ] .

Solution Summary: The author explains how to obtain the expressions for E(X) right and V left (X-mu

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Obtain the expression for E[g(X)] and V[g(X)].

Calculate the approximations to μR and σR.

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Chapter 4 Solutions