Sampling of Random Variables and Simulation

In this chapter we introduce the art of sampling of random variables. Sampling a random variable X, for example real valued, means using a random number generator to generate n real numbers \((x_1,\ldots ,x_n)\), realizations of a sample \((X_1,\ldots ,X_n)\) of independent and identically distributed random variables sharing the same law as X. The ability of sampling is crucial to deal with integrals in high dimensions, appearing in quantum mechanics and in statistical physics, and gives the possibility to simulate physical systems. We first introduce simple tools to sample random variables, that can be used only in quite special situations. In the last part of the chapter, we will introduce and discuss a very general sampling technique, relying on the Metropolis theorem.