Analogous to poisson distribution, but for continuous random variable. Consider a distribution which lasts a duration of time until success; what’s the probability that success is found in some range of times:

“What’s the probability that there are an earthquake in \(k\) years if there’s on average \(2\) earthquakes in 1 year?”

## constituents

- $λ$—“rate”: event rate (mean occurrence per time)

## requirements

\begin{equation} f(x) = \begin{cases} \lambda e^{-\lambda x}, x\geq 0\\ 0, x< 0 \end{cases} \end{equation}

## additional information

**expectation**: \(\frac{1}{\lambda}\)**variance**: \(\frac{1}{\lambda^{2}}\)

### exponential distribution is memoryless

An exponential distribution doesn’t care about what happened before.

“On average, we have a request every 5 minutes. There have been 2 minutes with no requests. What’s the probability that the next request is in 10 minutes?”

is the same statement as

“On average, we have a request every 5 minutes. ~~There have been 2 minutes with no requests.~~ What’s the probability that the next request is in 10 minutes?”

That is:

\begin{equation} P(s+t|s) = P(t) \end{equation}