probability mass and density functions

Once you have a random variable, you need a way to say how the probability is spread across its possible values. For a discrete variable this is the probability mass function (PMF): it simply states the probability of each individual value, like 'P(X = 0) = 0.9, P(X = 1) = 0.09, ...'. For a continuous variable it is the probability density function (PDF), a smooth curve whose height shows where values cluster, with area — not height — giving probability.

The PMF, often written p(x) or f(x), gives an honest probability for each value, and all those probabilities add up to 1. The PDF, written f(x), is subtler: its value at a point is not a probability but a density (probability per unit of x). To get an actual probability you take the area under the curve between two points, and the total area under the whole curve is 1. A density can rise above 1 in places — that is fine, because it is a rate, not a probability; only its accumulated area is capped at 1. Where the curve is tall, values are more likely to fall nearby.

These two functions are the working description of every distribution an actuary uses. The bell-shaped normal PDF, the long right tail of a lognormal severity PDF, the spiky PMF of a Poisson claim count — each is a picture of where risk concentrates. Reading a density's shape tells you at a glance whether losses are tightly clustered or prone to rare giants, which is exactly the information pricing and capital decisions hinge on.