In my last post I derived the formula for the expected maximum of *n* independent and identically distributed (iid) exponential random variables. If each random variable has rate parameter , then the expected maximum is , where is the *n*th harmonic number.

This post considers the discrete version of the same problem: the expected maximum of *n* independent and identically distributed geometric random variables. Unfortunately, there is no known nice, closed-form expression for the expected maximum in the discrete case. However, the answer for the corresponding exponential random variables turns out to be a good approximation.

Let be *n* iid geometric(*p*) random variables. Let denote the maximum of the variables. Let . Let the rate parameter for the corresponding exponential distribution be . Now, we have

as the cumulative distribution function of a geometric(*p*) random variable is .

By viewing the infinite sum as right- and left-hand Riemann sum approximations of the corresponding integral we obtain

Now, let’s take a closer look at the integral that occurs on both sides of this inequality. With the variable switch we have

proving the fairly tight bounds

We can obtain a more precise approximation by using the Euler-Maclaurin summation formula for approximating a sum by an integral. Up to a first-order error term, it says that

yielding the approximation

with error term given by

One can verify that this is quite small unless *n* is also small or *q* is extreme.

See also Bennett Eisenberg’s paper “On the expectation of the maximum of IID geometric random variables,” *Statistics and Probability Letters* 78 (2008), 135-143.

(I first posted this argument online as my answer to “Expectation of the maximum of IID geometric random variables” on Math Stack Exchange.)

Really enjoyed this and the similar article on the exponential. Do you have a similar derivation for the expected maximum of iid Poisson rvs?

Thanks! I don’t have a corresponding derivation for Poisson random variables, and I don’t think I’ve ever seen one, either. An interesting idea for a future post, perhaps.