 0.45, 7.92, negative 0.06, and the list goes on forever. These numbers may not seem unusual at first glance, but here are some facts that may surprise you. This set of numbers has no expectation. This set of numbers also has no variance. This set of numbers never converges. Welcome to the Cauchy distribution. I'm going to use some computer simulation to compare standard distributions to the Cauchy. Here are a few convergence plots. Each of these panes represent samples from different distributions. The x-axis represents the number of draws made from a distribution. The y-axis represents the mean of those samples. I iterated 50 times for each sample size. That is why there's multiple lines. On the left, I have the shape of the distributions that generated these convergence plots. All the convergence plots look very similar. They start wide, then taper down as the number of samples increase. What I think is wild is that the distributions I pulled from to generate these plots look very different. This phenomenon is known as the central limit theorem. Large enough samples from any distribution will end up looking like a normal distribution. The exception is the Cauchy distribution. The Cauchy distribution is a pathological distribution. Where other distributions have a defined expectation, the Cauchy does not. It refuses to obey or conform to the central limit theorem and will never converge no matter how large the samples. I'm going to go into the math a little here. A practical definition of convergence requires three things. A sample mean, a population mean, a threshold. Here is a rough definition of convergence. As a larger sample is taken, then means of those larger samples are nearly guaranteed to be smaller than some arbitrarily small threshold. Here is a visual representation of this definition. For the uniform distribution, here are a plot of different thresholds. It is clear that for some of these thresholds, the probability is going to be one. As the threshold gets smaller, more samples are needed to have the assurance that the sample mean will meet the population mean. This is how things should look. This is how things do look for all distributions. But this is how the Cauchy distribution compares. The probability that the difference in the sample mean and the population is less than the threshold is never one. And grabbing more samples doesn't change the story. The numbers I showed at the beginning of the video are draws from the Cauchy distribution. Without a convergence in probability, it is impossible to take an average. I mean, you can average the sample, but it says nothing about the population average. So what are the most stubborn numbers? The infinite string of values generated from the Cauchy distribution. Thanks for watching.