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).
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
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,