The Exponential Distribution

The exponential distribution is simply the standard model for how long you wait until the next random event, like the next phone call or the next radioactive decay. It's named for the exponential function $e^{-\lambda x}$ that drives it. The formulas look scary with that $e$ in them, but there are only three pieces. $f(x) = \lambda e^{-\lambda x}$ (the density, highest at $x = 0$ and fading away), $P(X > x) = e^{-\lambda x}$ (the chance you're still waiting after time $x$, which shrinks steadily) and $\mu = \frac1\lambda$ (if 4 calls come per hour, you wait a quarter hour on average). That's it; the rest of this lesson is what follows from those three.

Link to Poisson: you've already met the Poisson distribution, which counts how many events happen in a stretch of time. The exponential is the other side of the same coin: if events arrive as a Poisson process with rate $\lambda$, the times between them are exponential with mean $\frac1\lambda$. Here's why the survival formula looks the way it does. Still waiting at time $x$ means zero events in $[0, x]$, and the Poisson chance of zero events with mean $\lambda x$ is $e^{-\lambda x}$.

Memoryless property: $P(X > s + t \mid X > s) = P(X > t)$. Think about why. Events don't know how long it's been since the last one; the rate is the same $\lambda$ at every moment. So if you've waited $s$ minutes with nothing happening, the future looks exactly like it did at the start. You can check it with the formula: $\frac{e^{-\lambda(s + t)}}{e^{-\lambda s}} = e^{-\lambda t}$. A light bulb that has burned 1,000 hours is, under this model, as good as new.

The median solves $e^{-\lambda x} = \frac12$: $x = \frac{\ln 2}{\lambda}$, less than the mean because of the long right tail.

A common mistake. It's tempting to think a long wait makes the event "due," the way people say an earthquake is overdue. Under the exponential model it isn't. Having waited already doesn't shorten the wait ahead at all. (Things that wear out, like brake pads, don't follow this model, and we'll need other distributions for them.)

Time between customer arrivals, radioactive decays and server requests are modeled as exponential; reliability engineers use it for components that fail at random rather than by wearing out.