4 Let X be a random variable having finite expectation and variance , and let g( ) be a twice differentiable function Show that $$E g(X) g() frac g ' '() 2 $$ HINT Expand g( ) in a Taylor series about Use the first three terms and ignore the remainder ...

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4. Let X be a random variable having finite expectation and variance , and let g(-)...

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4. Let X be a random variable having finite expectation µ and variance σ², and let g(-) be a twice differentiable function. Show that

$$E[g(X)] = g(µ) + \frac{g''(µ)}{2} σ²$$

HINT: Expand g(-) in a Taylor series about µ. Use the first three terms and ignore the remainder.

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A First Course In Probability

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Authors: Sheldon M. Ross

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4. Let X be a random variable having finite expectation and variance , and let g(-)...

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