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)]

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.