---
title: "exponential distribution"
type: concept
related: [Exponential Distribution, Random Variables, Probability, Probability Of K In X Time]
source: https://www.jemoka.com/posts/kbhexponential_distribution/
confidence: high
status: active
---

Analogous to poisson distribution, but for continuous random variable. Consider a distribution which lasts a duration of time until success; what&rsquo;s the probability that success is found in some range of times:
&ldquo;What&rsquo;s the probability that there are an earthquake in \(k\) years if there&rsquo;s on average \(2\) earthquakes in 1 year?&rdquo;
constituents $λ$&mdash;&ldquo;rate&rdquo;: event rate (mean occurrence per time) requirements \begin{equation} f(x) = \begin{cases} \lambda e^{-\lambda x}, x\geq 0\\ 0, x&lt; 0 \end{cases} \end{equation}
additional information expectation: \(\frac{1}{\lambda}\) variance: \(\frac{1}{\lambda^{2}}\) exponential distribution is memoryless An exponential distribution doesn&rsquo;t care about what happened before.
&ldquo;On average, we have a request every 5 minutes. There have been 2 minutes with no requests. What&rsquo;s the probability that the next request is in 10 minutes?&rdquo;
is the same statement as
&ldquo;On average, we have a request every 5 minutes. There have been 2 minutes with no requests. What&rsquo;s the probability that the next request is in 10 minutes?&rdquo;
That is:
\begin{equation} P(s+t|s) = P(t) \end{equation}
