---
title: "Binomial distribution"
output: html_notebook
---

This code illustrates properties of the binomial distribution in the context of genetic drift. `p_0` is the allele frequency in the parental population, `n_e` is the effective size. Notice that the median outcome is the same as `p_0`, indicating that half of the time the allele frequency is less than or equal to `p_0` and half the time it is greater than or equal to `p_0`.

Try some different values of `p_0` to convince yourself this is true. 

```{r}
options(tidyverse.quiet = TRUE)
library(tidyverse)
library(ggplot2)

p_0 <- 0.5
n_e <- 100

## Take 10000 samples from a binomial distribution with number of gametes
## equal to 2 times the effective population size and allele frequency p_0
##
dat <- tibble(p = rbinom(10000, 2*n_e, p_0)/(2*n_e))
median <- quantile(dat$p, c(0.5))
title <- sprintf("Median outcome: %4.2f", median)
p <- ggplot(dat, aes(x = p)) +
  geom_histogram(binwidth = 1/(2*n_e)) +
  geom_vline(xintercept = p_0, color = "salmon", linetype = "dashed") +
  xlab("p") +
  ggtitle(title) + 
  theme_bw()
p
```