The accurate estimation of probabilities of rare events through fast simulation is a primary concern of importance sampling. Rare events are almost always defined on the tails of probability density functions. They have small probabilities and occur infrequently in real applications or in a simulation. This makes it difficult to generate them in sufficiently large numbers that statistically significant conclusions may be drawn. However, these events can be made to occur more often by deliberately introducing changes in the probability distributions that govern their behavior. Results obtained from such simulations are then altered to compensate for or undo the effects of these changes. In this chapter the concept of IS is motivated by examining the estimation of tail probabilities. It is a problem frequently encountered in applications and forms a good starting point for the study of IS theory.
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