poisson distribution

Let’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.
“What’s the probability that there are k earthquakes in the 1 year if there’s on average 2 earthquakes in 1 year?”
where:
events have to be independent probability of sucess in each trial doesn’t vary constituents λ—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:
“what’s the probability of large n number samples getting k events with probability of \frac{\lambda}{n} of events”

\begin{equation} P(X=k) = \lim_{n \to \infty} {n \choose k} \left(\frac{\lambda}{n}\right)^{k}\left(1- \frac{\lambda}{n}\right)^{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’s just the sample mean