What a probability distribution is

A single number is a bad way to describe an uncertain future, and a distribution is the fix. It is simply the full inventory: every outcome that could occur, with a weight attached saying how much of the total likelihood it receives.

Two properties define one. All the weights are non-negative, and they sum to exactly one. That is the entire formal requirement. Everything else people say about distributions — averages, standard deviations, percentiles — are summaries computed from the weights, not additional assumptions.

The reason to work with distributions rather than point estimates is that different distributions with the same average behave completely differently. Consider two investments, each with an expected return of eight percent. The first returns between six and ten percent in every scenario. The second returns twelve percent in nine cases out of ten and loses twenty-eight percent in the tenth. Identical averages, and no one would call them the same thing.

That difference is invisible in the average and obvious in the distribution, and it is precisely the difference that matters to someone with borrowed money. The first investment never reaches a level that triggers anything. The second one does, once in ten.

The rest of this track is about which summaries of a distribution are useful, which are misleading, and how to generate a distribution when you cannot derive one.