A P&L is a number. A default count is a number. The number of limit orders filled in the next second is a number. None of the Probability Foundations topic's machinery — sample spaces, events, conditioning — directly hands you a number; it hands you probabilities of abstract sets. A random variable closes that gap: it is a numerical summary of the experiment's outcome, and its probability mass function (PMF) is the complete numerical description of how likely each value is. This note defines both, shows how to compute a PMF mechanically from the underlying sample space, and introduces the four named discrete distributions — Bernoulli, Binomial, Geometric, Poisson — that recur throughout the rest of quantitative finance as the building blocks of default indicators, order counts, and arrival processes.
Learning objectives. By the end of this note you should be able to:
- define a (discrete) random variable as a function on the sample space and state what its PMF captures;
- compute a PMF mechanically by collecting outcomes and summing their probabilities;
- state the PMF of the Bernoulli, Binomial, Geometric, and Poisson distributions and identify which experiment each models;
- verify that a candidate PMF is legitimate via the normalisation check .
In this lesson
- Random variables
- The probability mass function
- The named discrete distributions
Before you start
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