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  "body": "\n## Definition\n\nLet $\\lambda \u003e 0$.\nThe Poisson distribution with parameter $\\lambda$ is the distribution on $\\{0, 1, 2, \\dots\\}$ with the [[probability mass function|0WF4EJP91W]]\n$$p(k) = \\frac{\\lambda^k e^{-\\lambda}}{k!} \\quad \\text{for } k = 0, 1, 2, \\dots$$\nHere $k!$ is the [[factorial|0WFCGVMRRC]] of $k$. One writes $X \\sim \\operatorname{Poisson}(\\lambda)$.\n\n## Remarks\n\nThe values add up to $1$ because $\\sum_{k \\ge 0} \\lambda^k / k! = e^{\\lambda}$.\nThe mean and the variance are both $\\lambda$.\nIt is the limit of the [[binomial distribution|0VRWZDWW6B]] with parameters $n$ and $\\lambda / n$ as $n \\to \\infty$. This is the [[Poisson limit theorem|0XKTZRWTXX]].\nSiméon Denis Poisson introduced the distribution in 1837.\n\n## Explanations\n\n### L1 human\n\nThe Poisson distribution counts rare events that happen at random, such as calls to a help desk in one hour.\nIf $\\lambda = 2$ calls come per hour on average, the chance of no call in an hour is $e^{-2} \\approx 0.135$.\n",
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