Poisson Distribution_HA_JMKII.Rmd

---
title: "Poisson Distribution"
author: "Jan Kärstens"
date: "07.12.2018"
output: html_document
---

```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE)
```
We define a matrix "ma" in the range from 1 to "maxi" with "n" values.

```{r cars}
#Define desired range of matrix values (1 to maxi)
maxi <- 100
#Define desired size of matrix in range of 1 to maxi
n <- 10

#Matrix ma defined with 10 elements in the range of 1 to 100; these elements are part of the events in the range of lambda
ma<- sample(1:maxi, n)
returnValue(ma)

```

Next we define a range of all events "lambda" in which our matrix is included, here set 1 to 200 and define its Prior "prior_p". The prior is defined as 1 / Sum of ma. This is done for all events in lambda.

```{r}
#Range of all events (including ma) defined
lambda <- 1:200

#prior probability p(lambda) defined for all events of lambda (1-200)
prior_p <-rep(1 / sum(ma), length(lambda))

```

We define another Matrix "l_pois" in the length of lambda. In the matrix we store the product of the poisson distribution of the 10 events stored in Matrix "ma".
```{r}
#"Container" (i.e. Matrix) to store results of product of poisson distributions
l_pois <- rep(NA, length(lambda))

#loop in length of prior_lambda (1:200) to form product of poisson distributions
for (i in seq_along(lambda)){
  #product of poisson distr. at element i generated and stored in container at position i
  l_pois[i] <- prod(dpois(ma, i)) 
}
```
Finally we calculate the wanted Posterior "bayesnum" and Plot it. And receive the wanted Poisson Distribution with a Gaussian-like bell shape.
```{r}
#Calculating Bayes law and plotting values
bayesnum <- l_pois * prior_p
bayesnum <- bayesnum / sum(bayesnum)
plot(bayesnum, type = "h", col = "chocolate")
```