# Poisson Distribution

> ***GitHub***: https://github.com/czs108

## Definition

\begin{equation}
P(X = r) = \frac{e^{-\lambda} \cdot {\lambda}^{r}}{r!}
\end{equation}

\begin{equation}
\lambda = \text{The mean number of occurrences in the interval or the rate of occurrence.}
\end{equation}

If a variable $X$ follows a *Poisson Distribution* where the *mean* number of occurrences in the interval or the rate of occurrence is $\lambda$. This can be written as

\begin{equation}
X \sim Po(\lambda)
\end{equation}

## Expectation

\begin{equation}
E(X) = \lambda
\end{equation}

## Variance

\begin{equation}
Var(X) = \lambda
\end{equation}

## Approximation

A *Binomial Distribution* $X \sim B(n,\, p)$ can be approximated by $X \sim Po(np)$ if $n$ is *large* and $p$ is *small*.

Because both the *expectation* and *variance* of $X \sim Po(np)$ are $np$. When $n$ is large and $p$ is small, $(1 - p) \approx 1$ and for $X \sim B(n,\, p)$:

\begin{equation}
E(X) = np
\end{equation}

\begin{equation}
Var(X) \approx np
\end{equation}

The approximation is typically very close if $n > 50$ and $p < 0.1$.

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exercises/Poisson Distribution.ipynb
czs108/Probability-Theory-Exercises
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exercises/Poisson Distribution.ipynb
czs108/Probability-Theory-Exercises
60c6546db1e7f075b311d1e59b0afc3a13d93229
MIT
exercises/Poisson Distribution.ipynb
czs108/Probability-Theory-Exercises
60c6546db1e7f075b311d1e59b0afc3a13d93229
MIT
2022-03-21T05:04:07.000Z
2022-03-21T05:04:07.000Z
Qwen/Qwen-72B
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