Beautiful Testing

Nonuniform Random Number Generators

Suppose you want to decide how many cashiers to hire for a grocery store. If shoppers arrived at regular intervals and all took the same amount of time to check out, this would be a simple problem. But in reality, shoppers do not arrive like clockwork. Nor do they all take the same amount of time to check out. Some shoppers dart into the store for a pack of gum, whereas others come to stock up on a week’s worth of provisions for a family of eight.

There is an entire branch of mathematics called queuing theory devoted to studying problems like how many cashiers a grocery store needs. Often, queuing theory assumes that the time needed to serve a customer is exponentially distributed. That is, the distribution of service times looks like the function e^−x. A lot of customers are quick to check out, some take a little longer, and a few take very long. There’s no theoretical limit to how long service may take, but the probability decreases rapidly as the length of time increases, as shown in the first image that follows. The same distribution is often used to model the times between customer arrivals.

The exponential distribution is a common example of a nonuniform distribution. Another common example is the Gaussian or “normal” distribution.^[55] The normal distribution provides a good model for many situations: estimating measurement errors, describing the heights of Australian men, predicting IQ test scores, etc. With a normal distribution, values tend ...