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@jemoka / Jemoka Knowledge Base / wiki/concepts/exponential_distribution.md
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--- 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’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}