---
title: "Poisson Params"
author: "Russell Almond"
date: "September 1, 2020"
output: html_document
runtime: shiny
---

```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE)
library(shiny)
library(ggplot2)
```

The Poisson distribution is a distribution for counts of events.  

Assume the following things:

* Events happen at a rate $\lambda$ per unit interval on average.

* Count the number of events in a time interval of $T$ units.

* Assume that the events happen at a uniform rate throughout the interval (e.g., we don't get more customers in the morning than the afternoon).

The the number of events, $X$, follows a _Poisson distribution_.

$$P(X=x) = \frac{(\lambda T)^x}{x!}e^{-\lambda T}$$
The distribution looks like:

```{r density, echo=FALSE}
inputPanel(
  sliderInput("mean", label = "Expected number of events per unit time",
              min=0, max=100, value=3.5, step=1),
  
  sliderInput("t", label = "Time Interval:",
              min = 0, max = 365, value = 1, step = 1)
)

renderPlot({
  mu <- as.numeric(input$mean) * as.numeric(input$t)
  n <- mu + 3* sqrt(mu)
  dat <- data.frame(x=0:n,y=dpois(0:n,mu))
  ggplot(dat,aes(x,y)) +geom_col()  

})
```

The mean and variance of the Poisson distribution are $\lambda T$ and $\lambda T$.  

As the variance grows pretty quickly, statisticians will often take the square root of count data (especially if there is heteroscedasticity) to stabilize the variance.