Cumulative distribution function

All distributions — discrete and continuous — can be described with a single tool: the cumulative distribution function, which accumulates the probability up to a given value.

Definition — Cumulative distribution function

The cumulative distribution function of $X$ is $F(x) = P(X\le x) = \int_{-\infty}^x f(t)\,dt.$ It is increasing and goes from $0$ to $1$. To compute the probability of an interval one uses the difference: $P(a\le X\le b) = F(b)-F(a).$

The practical advantage is enormous: once $F$ is known, every interval probability reduces to a subtraction. This is exactly the logic of the standard normal tables, where one reads $F$ and combines the values.

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Topics: Distribuzioni probabilita
Concepts: Densita di probabilita · Funzione di ripartizione
Skills: Usare formule