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  "body": "\n## Definition\n\nLet $r \\ge 1$ be an integer and let $p \\in (0, 1]$.\nThe negative binomial distribution with parameters $r$ and $p$ is the distribution on $\\{0, 1, 2, \\dots\\}$ with the [[probability mass function|0WF4EJP91W]]\n$$p(k) = \\binom{k + r - 1}{k} p^r (1 - p)^k \\quad \\text{for } k = 0, 1, 2, \\dots$$\n\n## Remarks\n\nIt is the distribution of the number of failures before the $r$-th success in independent Bernoulli trials with success probability $p$. The last trial is a success, and the $k$ failures are among the other $k + r - 1$ trials.\nFor $r = 1$ it is the [[geometric distribution|0XCVV5Q500]]. The sum of $r$ independent geometric random variables with parameter $p$ has the negative binomial distribution with parameters $r$ and $p$.\nThe mean is $r(1 - p)/p$ and the variance is $r(1 - p)/p^2$.\nSome authors count the trials, or the successes before a fixed number of failures. Check the convention of each source.\n\n## Explanations\n\n### L1 human\n\nToss a coin until you have seen $r$ heads, and count the tails on the way.\nThis count has the negative binomial distribution. For $r = 1$ you wait only for the first head.\n",
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