Poisson_Processes.Rmd

---
title: "Poisson Processes"
author: "Tom Caputo"
date: "Friday, September 25, 2015"
output: html_document
---
#####What is a Poisson Processes?
A poisson processes is a stochastic proceses that counts the arrival of a number of events in a given time interval. We use a poisson processes in situations where we a counting the occurances of a particular event that appear to happen a certain rate. For example:

- Number of times your car will break
- Number of users visiting a website
- Arrival times of train

These processes must be random (not following a deterministic structure). So pretty much, like all random variables there has to be an unpredictability to it.

#####Mathmatical Definition

Poisson processes a characterizzed by two parameters...

$\lambda$: The rate of occurance. (sometimes called the intensity)

$\tau$: The interval

...and follows a Poisson distribution with associated pearameters $\lambda\tau$.

$P[N(t + \tau) - N(t) = k] = \frac{e^{-\lambda\tau}(\lambda\tau)^k}{k!}$

Let's simulate some Count data

```{r,  warning=FALSE}
poisson_data <- rpois(n = 200, lambda = 10)

hist(poisson_data, main = "Poisson Data", xlab = "", breaks = 50)

#Variance = Mean in Poisson Distributed Data
mean(poisson_data)
var(poisson_data)
```

 As $\lambda$ approaches 30 and beyond, the distribution starts to look a lot like the Normal Distribution. This is because $Poisson(30)$ can be thought of the aggregate of 30 poisson variables and thus can be considered Normally distributed according to the Central Limit Theorem. Don't believe me? Look at these Graphs

```{r warning=FALSE, message = FALSE}
library(ggplot2)
pois.data <- rbind(
  data.frame(lambda = "lambda = 1", data.pois = rpois(n=200, lambda =1)),
  data.frame(lambda = "lambda = 5", data.pois = rpois(n=200, lambda =5)),
  data.frame(lambda = "lambda = 15", data.pois = rpois(n=200, lambda =15)),
  data.frame(lambda = "lambda = 30", data.pois = rpois(n=200, lambda =30))
)

ggplot(data = pois.data) +
  ggtitle("Poisson Distribution with Different Rates(Lambda)") +
  geom_density(aes(x = data.pois) ) +
#   scale_x_discrete(breaks = 1) +
  xlab("") +
  ylab("Instances") +
  facet_wrap(~lambda)
```

#####Non-Stationary Poisson Processes

Non-Stationary Poisson processes, sometimes called inhomogenous poisson process, is a poisson processes with a variable rate $\lambda$. This occurs often in time dependent processes. For example imagine a the rate a customer arrives a grocery store. Early morning the rate is low and increases throughout the day. As the time gets closer to midnight/closing time the rate customers arrives decreases. 

The rate $(\lambda)
$N_{a,b} = \int_{a}^{b} \lambda(\tau)d\tau$