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  "body": "\n## Definition\n\nLet $p \\in [0, 1]$.\nThe Bernoulli distribution with parameter $p$ is the distribution on $\\{0, 1\\}$ with the [[probability mass function|0WF4EJP91W]]\n$$p(1) = p, \\qquad p(0) = 1 - p.$$\nEquivalently, $p(k) = p^k (1 - p)^{1-k}$ for $k \\in \\{0, 1\\}$. One writes $X \\sim \\operatorname{Bernoulli}(p)$.\n\n## Remarks\n\nThe mean is $p$ and the variance is $p(1 - p)$.\nThe indicator $\\mathbf{1}_A$ of an event $A$ has the Bernoulli distribution with parameter $P(A)$.\nIt is the distribution of the outcome of one [[Bernoulli trial|0PHC9FE2V5]], with $1$ for success. It is the [[binomial distribution|0VRWZDWW6B]] with $n = 1$.\n\n## Explanations\n\n### L1 human\n\nThe Bernoulli distribution describes one yes-or-no experiment: $1$ means yes and $0$ means no.\nA fair coin, with $1$ for heads, has the Bernoulli distribution with $p = 1/2$.\n",
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