Poisson Distribution - lec. 13

ucla | MATH 170E | 2023-02-08T17:42

Definitions

there are $λ > 0$ occurrences
let $X$ be the number of occurrences in some time span and takes values from $S = 1, 2, \dots$ (assuming population is infinite)
Assuming time intervals are disjoint: $(t_{1}, t_{2}], (t_{2}, t_{3}], \dots, (t_{n}, t_{n + 1}] ⟹$ then occurrences on each time interval are independent
If $h = t_{2} - t_{1} > 0$ is sufficiently small $⟹ P (X = 1 in (t_{1} t_{2}]) = λ h$ and converges rapidly to zero as $h \to 0$ i.e. an approximate Poisson r.v.

$X \sim Poisson(λ)$

$p_{X} (x) = \frac{e^{- λ} λ^{x}}{x!}$

$F_{X} (x) = e^{- λ} \sum_{k = 0}^{x} \frac{λ^{k}}{k!}$

$\mathbb E[X]=\lambda$

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