The exponential

The exponential distribution is defined such that its growth rate at any instant is equal to its value. It’s also well known that . There’s a nice combinatorial way to connect these two definitions:

First, consider the discrete version of , . We can find by the following process: let be (that is, an empty set of sets). Then, define for to be . That is, we take every set from , then decide whether to add into it or not. Here a few values of :

We can see how this simulates a growth rate equal to value, since clearly, . Therefore, as desired. However, is also just the number of ways to select a subset of the integers in . How can we express this in a continuous fashion?

Loosely speaking, the “number of ways” to select numbers in the range should just be (since in the continuous case, there is 0 chance of picking two equal numbers). Therefore, summing over all possible values of , we arrive at our desired expression.

Let’s formalize “number of ways.” Note that instead of , we now want . This means we need to slightly redefine . Currently, it’s simply defined as the number of sets in , or , but instead we want

That is, sets are now assigned a “weight” which decreases as the set size increases. It can be shown that under this definition, the loose argument made above can be made precise.

This definition of can also be interpreted probabilistically by defining a random process in which each interval is selected independently with probability . This corresponds to a Poisson process with . In particular, the chance of selecting exactly intervals is just

as expected ( appears in the denominator of the LHS as a normalizing factor).