Monte Carlo simulation¶

Using the probability distribution the probability distribution of a random vector , we seek to evaluate the following probability:

Here, is a random vector, a deterministic vector, the function known as limit state function which enables the definition of the event .

If we have the set of N independent samples of the random vector , we can estimate as follows:

where describes the indicator function equal to 1 if and equal to 0 otherwise; the idea here is in fact to estimate the required probability by the proportion of cases, among the N samples of , for which the event occurs.

By the law of large numbers, we know that this estimation converges to the required value as the sample size N tends to infinity.

The Central Limit Theorem allows one to build an asymptotic confidence interval using the normal limit distribution as follows:

with and is the -quantile of the standard normal distribution.

One can also use a convergence indicator that is independent of the confidence level $alpha$: the coefficient of variation, which is the ratio between the asymptotic standard deviation of the estimate and its mean value. It is a relative measure of dispersion given by:

It must be emphasized that these results are asymptotic and as such needs that . The convergence to the standard normal distribution is dominated by the skewness of divided by the sample size N, it means . In the usual case , the distribution is highly skewed and this term is approximately equal to . A rule of thumb is that if with , the asymptotic nature of the Central Limit Theorem is not problematic.

The method is also referred to as Direct sampling, Crude Monte Carlo method, Classical Monte Carlo integration.