Sampling from heavy vs light tailed distribution I am having some issue understanding the behavior of such distributions when generating random numbers.
I was under the impression that heavy tailed distributions have "heavier" tails, so there is more probability to observe higher values, whereas lighter tailed distributions have values more concentrated in the body of the distribution. Is this correct?
I tried to sample from a Cauchy distribution (heavy distribution) and from a t-distribution (light) and plot the histogram. I am confused because I expected exactly the opposite of what I get. Here some example in R (the same results can be replicated with any statistical software)
set.seed(999)

heavy_data <- rcauchy(1000)
light_data <- rt(1000, 10)

hist(heavy_data)
hist(light_data)



It looks like that from the cauchy distributions, all the observations are in the body with almost anything in the tails, whereas for the t-distributions we have a wider spread of data, so in the body as well as in the tail.
Could anyone clarify this?
 A: The direct answer to the question is, no, heavier tails does not necessarily mean "more probability in the tails." A sequence of distributions can have increasing tail weight, with simultaneously less probability, as long as the tails extend further and further.
See here for an example.  https://math.stackexchange.com/a/2510884/472987
Part of the problem is that there are incorrect sources all over the web that show "fat tailed" distributions using histograms with a good chunk of probability in the tails.  The problem is that, as the OP notes, the tails, while thicker than the normal distribution, are still very close to zero and hence hard to visualize in a histogram.
Thus, histograms are not appropriate for visualizing fat tails. The normal quantile-quantile plot should be used instead. As it turns out, there is a very direct mathematical connection between kurtosis (a measure of fat/heavy tails) and the q-q plot, see here: https://stats.stackexchange.com/a/354076/102879
A: Your intuition is correct but your pictures are inaccurate. hist by default generates the limits of the x-axis based on the range of your data. Your Cauchy data ranges from about -400 to 400, whereas your t_10 data ranges from about -5 to 5. So you need to specify a common x-axis to compare. A related problem is the bin size. The bins of the Cauchy data are big, driven by the range of the data. A simple way to make them more comparable is to increase the number of bins:
hist(heavy_data, xlim = range(heavy_data), breaks = 600)
hist(light_data, xlim = range(heavy_data), breaks = 200)

