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  "body": "\n## Definition\n\nLet $\\lambda \u003e 0$.\nA homogeneous Poisson process with rate $\\lambda$ is a [[stochastic process|0QH5ENKZM1]] $(N(t))_{t \\ge 0}$ with values in $\\{0, 1, 2, \\dots\\}$ such that:\n1. $N(0) = 0$.\n2. For all $0 \\le t_0 \u003c t_1 \u003c \\dots \u003c t_n$, the increments $N(t_1) - N(t_0), \\dots, N(t_n) - N(t_{n-1})$ are [[independent|0XCE070V8B]].\n3. For all $0 \\le s \u003c t$, $N(t) - N(s)$ has the [[Poisson distribution|0VEFSSKX18]] with parameter $\\lambda (t - s)$.\n\n## Remarks\n\n$N(t)$ counts the points, or events, in the interval $(0, t]$. One usually also asks that the paths $t \\mapsto N(t)$ are right-continuous.\nThe times between consecutive events are i.i.d. with the [[exponential distribution|028Z8WC5Y9]] $\\operatorname{Exp}(\\lambda)$. Conversely, i.i.d. $\\operatorname{Exp}(\\lambda)$ gaps define a Poisson process with rate $\\lambda$.\nGiven $N(t) = n$, the $n$ event times in $(0, t]$ have the law of $n$ independent uniform points in $(0, t]$, put in increasing order.\nOn a general space $E$ with a measure $\\mu$, a Poisson point process has independent counts in disjoint sets.\nThe count in a set $B$ has the law $\\operatorname{Poisson}(\\mu(B))$ when $\\mu(B) \u003c \\infty$.\n\n## Explanations\n\n### L1 human\n\nThe Poisson process models events that happen \"completely at random\" in time, such as clicks of a Geiger counter or arrivals of customers.\nOn average, $\\lambda$ events happen per unit of time, and the waiting times between events are exponential.\n",
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