# Monte Carlo simulationΒΆ

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.

(`Source code`

, `png`

)

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