Monte Carlo simulation is a technique that uses random numbers and random variates to solve stochastic or deterministic problems that do not involve the passage of time. It is used to evaluate integrals of functions that cannot be directly integrated. The method involves defining a random variable equal to the function multiplied by the interval length and taking the sample mean of this random variable from running multiple simulations, which converges to the true expected value and integral.

1. Monte Carlo Simulation Adapted From Law and Kelton

2. What is Monte Carlo Simulation? A scheme employing random numbers, U(0,1) random variates, which is used to solving certain stochastic or deterministic problems where the passage of time does not play an important role. It is rather static than dynamic.

3. Example Ex. Suppose that we want to evaluate this integral: Where g(x) is a real valued function that is not integrable

4. To use simulation, let Y be that random variable (b-a)g(X), where X is a continuous random variable distributed uniformly on [a,b]. Thus, = I

5. But, E(Y) can be estimated using the concept of the sample mean . Thus,

6. Ex. Evaluate (Answer is 2). n 10 20 40 80 160 2.213 1.951 1.948 1.989 1.993