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  "body": "\n## Definition\n\nLet $p \\in [0, 1]$.\nA Bernoulli process with parameter $p$ is a [[stochastic process|0QH5ENKZM1]] $(X_n)_{n \\ge 1}$ of [[i.i.d.|0AJN7WHBJZ]] random variables with the [[Bernoulli distribution|0J97P5KZY0]] $\\operatorname{Bernoulli}(p)$:\n$$P(X_n = 1) = p \\quad \\text{and} \\quad P(X_n = 0) = 1 - p \\quad \\text{for each } n \\ge 1.$$\n\n## Remarks\n\nOne calls $X_n = 1$ a success and $X_n = 0$ a failure in trial $n$.\nThe number of successes in the first $n$ trials has the binomial distribution $\\operatorname{Bin}(n, p)$.\nFor $0 \u003c p \\le 1$, the number of failures before the first success has the geometric distribution with parameter $p$. The times between successes are i.i.d.\nThe Bernoulli process is a discrete-time analogue of the Poisson process.\n\n## Explanations\n\n### L1 human\n\nThe Bernoulli process is the mathematical model of tossing the same coin again and again, forever.\nEach toss is independent of the others, and each has the same probability $p$ of heads.\n",
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