Finding the expected maximum of independent and identically distributed (iid) exponentially distributed random variables is a standard calculation in an undergraduate course in probability theory. This post goes through the details of that calculation. (My next post will address the more difficult question of the expected maximum of iid geometric random variables.)
There are three properties of exponential random variables that we need.
- Exponentially distributed random variables are memoryless. Formally, if X is exponentially distributed, then . Less formally, suppose that the lifetime of an electronic component is exponentially distributed. The memoryless property says that the probability that the component will last at least t units longer, given that the component has lasted s units of time, is independent of the value of s.
- The minimum of n exponentially distributed random variables with rate parameters is itself exponentially distributed with rate parameter . (For a proof, see the Wikipedia page on the exponential distribution.)
- The exponential distribution is a continuous distribution. (This is a very basic property of the exponential distribution. However, the geometric distribution has the first two properties as well, yet its expected maximum is more complicated.)
Now, suppose we have n iid exponential() random variables, and let be the ith smallest of these. Since is exponential(, .
Since exponential random variables are continuous, the probability that any two of the n random variables have the same value is 0. Thus the n-1 other random variables all have values larger than . However, the memoryless property says knowledge of essentially “resets” the values of the other random variables, so that the time between and is the same (distributionally) as the time until the first of n-1 iid exponential() random variables takes on a value. Thus .
Applying the same logic, we get that, for , . Thus the expected value of the maximum of the n random variables is
where is the nth harmonic number.