Probability Distribution for the kth Greatest of a Sequence of n Random Numbers

This Demonstration shows a simple example of using extreme value theory to calculate the probability density function for the greatest number in a series of random numbers drawn from three distributions of some importance in financial calculations. Many observations of the greatest number in a sample of size drawn randomly from the specified distribution are taken and displayed via a histogram. The red curve is the graph of the analytical expression for the smallest number, derived by considering the probability that, in a sequence of random numbers, numbers are greater than and numbers are smaller, all this weighted by the binomial distribution (see Details).

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DETAILS

The probability that random numbers are smaller than is given by the joint probability that each one of the random numbers is smaller than . If the observations in the sample are independent of each other we then have
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where is the cumulative distribution function (CDF). The probability density function of the maximum of the observations is the derivative of . The generalization for the greatest number in a sequence is then given by the probability of obtaining in observations while obtaining observations which are greater than . To account for all possible permutations with which can be less than the probability must be weighted by a binomial factor,
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