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Expectation value of the maximum of two variables

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Consider n variables drawn independently from an arbitrary probability distribution. We derive the general expression of the probability density for the kth largest variable. Derive the general expression for the expectation value of the largest variable. Evaluate this expectation value for the standard normal distribution for the case n = 2.

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Given a probability density p(x) the probability density p(x_1,x_2,...,x_n) for n independent random variables x_1,x_2,...,x_n distributed according to p(x) is given by:

p(x_1,x_2,...,x_n) = p(x_1)p(x_2)...p(x_n) (1)

The probability density for the kth largest variable, r(k,x), can be obtained as follows. Of the n variables, k-1 are larger than or equal to x, n-k variables are smaller than or equal to x, and one variable is equal to x. We need to integrate p(x_1,x_2,...,x_n) over this allowed region to obtain r(k,x). We can do this by exploiting the permutation symmetry in (1), by ...

Solution Summary

In the solution everything is derived from first principles, finding the standard normal distribution.