\chapter{Introduction}
\label{ch:intro}


\section{Importance Sampling Estimator}

\subsection{Problem Statement}

The objective of this assignment is to estimate the right tail of an exponential random variable using ideas from importance sampling and tilted distributions.  

The problem to solve for this project is as follows:  Let $\mathbf{X}$ be an exponential random varibale with intensity $\lambda$ (mean $\frac{1}{\lambda})$.  For an constant $a>0$, the goal is to estimate the probability:
\begin{equation}
\theta = Pr\{X>a\} =\lambda \int^{\infty}_a e^{-\lambda\,x}dx
\end{equation}

Since the event $X > a$ occurs with a very small probability, i.e. 0 is very small, the
use of classical Monte Carlo approach will not be efficient. the usual approach would be completely inadequate since approximating $\mu$ to any reasonable degree of accuracy would require $n$ to be very large.  Using a much smaller value of $n$, would lead result in an estimate, $\hat{\theta} = 0$.  Instead, it is better to use importance sampling with a tilted density serving as the density to sample from.  In this paper, a method to reduce the variance of the esimator will be presented and numerical calcuations of accuarcy and computation time will be compared with classical approaches.




