Suppose we wish to estimate the probability of a rare event (such as a default probability). Let the random variable \(X\) be equal to 1 if the event occurs and to zero otherwise. Then \(p=\mathrm{E}[X]\). The standard Monte Carlo method takes \(n\) samples \(x_{i}\) and forms the average

\[\hat{p}=\frac{1}{n} \sum_{i=1}^{n} x_{i}\]

(a) What is the variance of \(\hat{p}\) ?

(b) What is the standard deviation of the absolute error \(\varepsilon_{\mathrm{ab}}=\hat{p}-p\) ?

(c) Define the standard deviation of the relative error as \(\varepsilon_{\mathrm{rel}}=\varepsilon_{\mathrm{ab}} / p\). Suppose \(p=1 \%\). About how many samples must be taken to reduce the relative error to \(1 \%\).