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  "body": "\n## Definition\n\nLet $p \\in (0, 1]$.\nThe geometric distribution with parameter $p$ is the distribution on $\\{0, 1, 2, \\dots\\}$ with the [[probability mass function|0WF4EJP91W]]\n$$p(k) = (1 - p)^k \\, p \\quad \\text{for } k = 0, 1, 2, \\dots$$\n\n## Remarks\n\nIt is the distribution of the number of failures before the first success in independent [[Bernoulli trials|0PHC9FE2V5]] with success probability $p$.\nThe other common convention counts the trials up to and including the first success. That number $Y = X + 1$ has $P(Y = k) = (1 - p)^{k-1} p$ for $k \\ge 1$.\nFor $X$ as in the definition, $P(X \\ge k) = (1 - p)^k$, the mean is $(1 - p)/p$ and the variance is $(1 - p)/p^2$.\nThe geometric distribution is the only distribution on $\\{0, 1, 2, \\dots\\}$ with the [[memoryless property|0ZAZZ34WQS]].\n\n## Explanations\n\n### L1 human\n\nToss a coin until the first head. The number of tails before it has the geometric distribution.\nEach extra tail multiplies the probability by $1 - p$, so the probabilities form a geometric sequence.\n",
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