Chapter 4
Numerical methods
4.1 Monte Carlo Method
Suppose we are given a random variable X and are interested in the evaluation of 
where g( · ) is some known function. If we are able to draw n pseudo random numbers x1, … , xn from the distribution of X, then we can think about approximating 
with the sample mean of the g(xi),
The expression (4.1) is not just symbolic but holds true in the sense of the law of large numbers whenever 
. Moreover, the central limit theorem guarantees that
where N(m, s2) denotes the distribution of the Gaussian random variable with expected value m and variance s2. In the end, the number we estimate with simulations will have a deviation from the true expected value 
of order 
. Given that P(|Z| < 1.96)0.95, Z N(0, 1), one can construct an interval for the ...
