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  "body": "\n## Definition\n\nLet $\\lambda \u003e 0$.\nThe exponential distribution with rate $\\lambda$ is the distribution on $\\mathbb{R}$ with the [[probability density function|04YG690N5V]]\n$$f(x) = \\begin{cases} \\lambda e^{-\\lambda x} \u0026 \\text{if } x \\ge 0, \\\\ 0 \u0026 \\text{if } x \u003c 0. \\end{cases}$$\nOne writes $X \\sim \\operatorname{Exp}(\\lambda)$.\n\n## Remarks\n\nThe cumulative distribution function is $F(x) = 1 - e^{-\\lambda x}$ for $x \\ge 0$, so $P(X \u003e x) = e^{-\\lambda x}$.\nThe mean is $1/\\lambda$, the variance is $1/\\lambda^2$ and the median is $\\ln 2 / \\lambda$. Some authors use the scale $\\beta = 1/\\lambda$ as the parameter.\nThe exponential distribution is [[memoryless|0ZAZZ34WQS]]: $P(X \u003e s + t \\mid X \u003e s) = P(X \u003e t)$ for $s, t \\ge 0$.\nIf $X_1, \\dots, X_n$ are independent with $X_i \\sim \\operatorname{Exp}(\\lambda_i)$, then $\\min_i X_i \\sim \\operatorname{Exp}(\\lambda_1 + \\dots + \\lambda_n)$.\n\n## Explanations\n\n### L1 human\n\nThe exponential distribution models the time until something happens when it can happen at any moment with the same intensity.\nExamples are the decay of a radioactive atom and the time until the next call to a help desk.\n",
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