(a) Suppose we have a function \(g\) of a random variable \(X\) whose PDF is \(f_{X}\). We wish to estimate \(\mathrm{E}(g(X))\). The first consideration is whether or not \(\mathrm{E}(g(X))\) exists. Let us assume that \(g\) is such that \(\mathrm{E}(g(X))\) exists and is finite. Another issue is whether we can generate random (or pseudorandom) variables that simulate observations on the random variable \(X\).

Given a pseudorandom sample \(x_{1}, \ldots, x_{m}\) from the distribution of \(X\), show that Monte Carlo estimate of \(\mathrm{E}(g(X))\),

\[ \widehat{\mathrm{E}(g(X)})=\frac{1}{m} \sum_{i=1}^{m} g\left(x_{i}\right) \]

is unbiased.

(b) Let \(X \sim \mathrm{N}(0,1)\). Use Monte Carlo to estimate \(\mathrm{E}\left(X^{2}\right)\).

(c) Use Monte Carlo simulation to estimate the conditional mean of a t random variable with \(4 \mathrm{df}\) given that the variable is greater than or equal to its \(95^{\text {th }}\) percentile.

i. In this case, you may assume that the \(95^{\text {th }}\) percentile is known. What other issues should you consider in reporting your result?

ii. For a general problem of this nature, we may not know the \(95^{\text {th }}\) percentile of the underlying distribution. If that is the case, how would you proceed?