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1 Introduction to Monte Carlo Methods     continued...

Statistical simulation methods may be contrasted to conventional numerical discretization methods, which typically are applied to ordinary or partial differential equations that describe some underlying physical or mathematical system. In many applications of Monte Carlo, the physical process is simulated directly, and there is no need to even write down the differential equations that describe the behavior of the system. The only requirement is that the physical (or mathematical) system be described by probability density functions (pdf's), which will be discussed in more detail later in this chapter. For now, we will assume that the behavior of a system can be described by pdf's. Once the pdf's are known, the Monte Carlo simulation can proceed by random sampling from the pdf's. Many simulations are then performed (multiple ``trials'' or ``histories'') and the desired result is taken as an average over the number of observations (which may be a single observation or perhaps millions of observations). In many practical applications, one can predict the statistical error (the ``variance'') in this average result, and hence an estimate of the number of Monte Carlo trials that are needed to achieve a given error.