The result of any estimation is a random variable. Suppose that we want to estimate the expectation E[X] of a random variable X via simulation. Most software programs have built-in random number generators, which approximately generate uniform random variables in (0,1). A simulation is a transformation of a sequence of random numbers into a random variable Y whose distribution reproduces or approximates that of X.
PROBLEM: Alas! The estimator Y is in general a random variable. Therefore, each time that we replicate our simulation, we produce a different instance of the estimator Y. In other words, our estimation has a prediction error.
SOLUTION: As with most numerical methods, we shall often be satisfied with an approximation, provided that we can control its precision. Statistical analysis can be used to estimate the error, which in the context of random variables has the form of a confidence interval.
The main problem with the method above is that we generally don't know the distribution F(y) of the estimators. The most common approach to estimate the errors is to use the Central Limit Theorem, when applicable.
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