Approximating Geometric Distributions with the exponential distribution

library(tidyverse)

Here is some code that you can use to explore how similar the exponential distribution is to the geometric distribution. Recall that it is easy to show in a Wright Fisher population of \(N\) diploids (i.e., with \(2N\) gene copies), that the probability that a pair of genes have a common ancestor \(t\) generations in the past is \[ \biggl(1 - \frac{1}{2N}\biggr)^{t-1} \frac{1}{2N} \] Let’s make a function that computes that for \(N\) and different values of \(t\):

G <- function(N, t) {
  (1 - (1 / (2 * N)))^(t - 1) * (1 / (2 * N) )
}

Then let’s look at those values for, say \(t=1,\ldots,50\) when \(N=30\):

geomet <- tibble(
  t = 1:50,
  prob = G(30, 1:50)
)

ggplot(geomet, aes(x = t, y = prob)) +
  geom_col(colour = "black", fill = "white", linewidth = 0.2) +
  theme_bw()

That defines the geometric distribution, which takes discrete values on \(1,2,\ldots\).

Now, remember how we said that a reasonable approximation to those values can be found with the exponential distribution, which also allows for real numbers. Here is a function that defines the approximating exponential distribution: \[ \frac{1}{2N} e^{-\frac{t}{2N}} \]

E <- function(N, t) {
  (1 / (2 * N)) * exp(-t / (2 * N))
}

And here we plot it in blue on top of the geometric distribution for \(N=30\):

expy <- tibble(
  t = 1:50,
  density = E(30, 1:50)
)

ggplot() +
  geom_col(data = geomet, aes(x = t, y = prob), 
           colour = "black", fill = "white", linewidth = 0.2) +
  geom_line(data = expy, aes(x = t, y = density), colour = "blue") +
  theme_bw()

That is pretty darn close, even for an \(N\) as small as 30. And the approximation gets better as \(N\) gets larger….