Probability theory

Most introductions to probability theory treat discrete probability distributions and continuous probability distributions separately. The measure theory-based treatment of probability covers the discrete, continuous, a mix of the two, and more.

So, the probability of the entire sample space is 1, and the probability of the null event is 0.

Continuous probability theory deals with events that occur in a continuous sample space.

In case the probability density function exists, this can be written as

In probability theory, there are several notions of convergence for random variables. They are listed below in the order of strength, i.e., any subsequent notion of convergence in the list implies convergence according to all of the preceding notions.

As the names indicate, weak convergence is weaker than strong convergence. In fact, strong convergence implies convergence in probability, and convergence in probability implies weak convergence. The reverse statements are not always true.

The first major treatise blending calculus with probability theory, originally in French: Théorie Analytique des Probabilités.
An English translation by Nathan Morrison appeared under the title Foundations of the Theory of Probability (Chelsea, New York) in 1950, with a second edition in 1956.