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I’m trying to get an intuition for gamma distributions, and why they are the model of choice for waiting times. In addition, I’d love to hear about any other distributions that are useful for modeling wait times for the same reason.

I’d say I’ve got a pretty solid understanding of why everyday random variables tend to follow a normal distribution. If you drop balls into a line of tubes from above, Galton board style, you’ll tend to see them arrange into a normal distribution. Add pegs to provide something for the balls to bounce off of, allowing small deviations in initial position to translate into larger deviations in which tube the balls end up in, and the standard deviation increases. This makes intuitive sense, and establishes very good intuitions around what types of random variables will tend to be normally distributed, and why.

Is there a way to explain the gamma distribution’s connection to wait times, an empirical argument, example, or combination of the two that makes it clear why wait times will tend to look like a gamma distribution? Further, are there other distributions that will tend to model wait times well for reasons that can be understood from a similar explanation?

Galton board Image courtesy of Seeking Wisdom.

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    $\begingroup$ Two reasons: (a) it has a nice shape with support on the positive real numbers and behaves nicely, and (b) the sum of $n$ iid exponentially distributed random variables has a gamma distribution, with exponential distributions representing gaps in a memoryless Poisson process so often assumed in modelling queues. $\endgroup$ Commented Jun 30, 2021 at 9:45
  • $\begingroup$ You can use moment generating functions to show that the sum of two indep exponential dist'ns with same rate is gamma dist'n with shape parameter 2. Etc. $\endgroup$ Commented Jun 30, 2021 at 17:32

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Your pictured device to illustrate a normal distribution is doing a mechanical simulation.

You can use a computer simulation in R to visualize the gamma distribution that results when three independent exponential random variables (each with mean 1) are added. Histogram of 100,000 sums of three exponential random variables:

enter image description here

R code for simulation and figure:

s = replicate(10^5, sum(rexp(3)))
hdr = "GAMMA(3, 1) Sum of Three Exponential RVs"
hist(s, prob=T, br=30, col="skyblue2", main=hdr)
curve(dgamma(x, 3, 1), add=T, col="maroon")
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