Expectation and Variance of Y = g(Z)

Question

Let $Z$ be a random variable with mean $E(Z)$ and variance $Var(Z)$. Consider the random variable given by $Y = g(Z)$, where $g$ is a monotonic and differentiable function over the domain of $Z$.

1. Write down the 1st-order Taylor series expansion of $g$ about $E(Z)$.

2. Using part 1, show that: $$E(Y)=g(E(Z))$$ $$Var(Y)=g′(E(Z))^2\cdot Var(Z)$$

For part $1$. I have that the Taylor series is given by $$g(E(Z)) + g'(E(Z))\cdot(Y - E(Z))$$

Is this correct?

I have used the fact that the first order Taylor series expansion is given by $$f(x) = f(a) + f'(a)\cdot(x-a)$$

I am unsure how to do the second part.

Answer 1

Your expansion is not quite correct.

$$ g(Z) = g(E[Z]) + g^\prime(E[Z])(Z - E[Z]) + O((Z-E[Z])^2)$$

By using linearity of expectation, the linear term should disappear. For the variance, re-arrange and square both sides to get,

$$ (g(Z) - g(E[Z]))^2 = g^\prime(E[Z])^2(Z - E[Z])^2 + O((Z-E[Z])^4)$$

Now take expectations.