When one tail of a distribution is unbounded, that is it has no theoretical maximum (or minimum). There are distributions where the tail can be quite long – there may be a small probability of very large or very small values, but it is still finite: for example, if you look at the number of double sixes when rolling two dice 100 times, the average is around 3, but there is a very small chance of there being a full 100 double sixes. However, there can never be more than 100. In the end the tail is bounded. In contrast, consider just starting tossing a coin until you reach the first tail and count how many heads you get before the first tail – that is a Negative Binomial distribution. Here the average value is just 1, but there is no maximum, you could, in principle, go on tossing heads for ever. That is the Negative Binomial distribution has an unbounded tail.
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