Suggest edit — poisson distribution

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---
title: "poisson distribution"
type: concept
related: [Probability Mass Function, Variance, Binomial Distribution, Probability, Probability Of K In X Time]
source: https://www.jemoka.com/posts/kbhprobability_of_k_in_x_time/
confidence: high
status: active
---

Let&rsquo;s say we want to know what is the chance of having an event occurring $k$ times in a unit time, on average, this event happens at a rate of $\lambda$ per unit time.
&ldquo;What&rsquo;s the probability that there are $k$ earthquakes in the 1 year if there&rsquo;s on average $2$ earthquakes in 1 year?&rdquo;
where:
events have to be independent probability of sucess in each trial doesn&rsquo;t vary constituents $λ$&mdash;count of events per time $X \sim Poi(\lambda)$ requirements the probability mass function:
\begin{equation} P(X=k) = e^{-\lambda} \frac{\lambda^{k}}{k!} \end{equation}
additional information properties of poisson distribution expected value: $\lambda$ variance: $\lambda$ derivation We divide the event into infinitely small buckets and plug into a binomial distribution, to formulate the question:
&ldquo;what&rsquo;s the probability of large $n$ number samples getting $k$ events with probability of $\frac{\lambda}{n}$ of events&rdquo;
\begin{equation} P(X=k) = \lim_{n \to \infty} {n \choose k} \qty(\frac{\lambda}{n})^{k}\qty(1- \frac{\lambda}{n})^{n-k} \end{equation}
and then do algebra.
And because of this, when you have a large $n$ for your binomial distribution, you can just use a poisson distribution, where $\lambda = np$.
adding poisson distribution For independent $A, B$
\begin{equation} A+B \sim Poi(\lambda_{A}+ \lambda_{B}) \end{equation}
MLE for poisson distribution \begin{equation} \lambda = \frac{1}{n} \sum_{i=1}^{n} x_{i} \end{equation}
yes, that&rsquo;s just the sample mean