The Poisson Distribution

x <- rpois(1000, 2)
hist(x)

Generating a Spatial Poisson Process

library(ggplot2)
library(dplyr)

## 
## Attaching package: 'dplyr'

## The following objects are masked from 'package:stats':
## 
##     filter, lag

## The following objects are masked from 'package:base':
## 
##     intersect, setdiff, setequal, union

x <- runif(100)
y <- runif(100)
df <- data.frame(x, y)

ggplot(df, aes(x = x, y = y)) + 
  geom_point()

df <- df %>%
  mutate(inside = ifelse(x < .75 & x > .25 & y > .25 & y < .75, TRUE, FALSE))
square <- data.frame(x = c(.25, .75, .75, .25), y = c(.25, .25, .75, .75))

ggplot(df, aes(x = x, y = y)) +
  geom_polygon(data = square, aes(x = x, y = y)) +
  geom_point(data = df, aes(color = inside)) +
  scale_colour_discrete(guide = FALSE)

simlist <- rep(NA, 1000)

for(i in 1:1000) {
  x <- runif(100)
  y <- runif(100)
  simlist[i] <- sum(x < .75 & x > .25 & y > .25 & y < .75)
}

hist(simlist)

plot(dpois(1:40, lambda = 100*.25))