Mathematical Explanation

The axioms of Kolmogorov.

Let S denote an event set with a probability measure P defined over it, such that probability of any event A ⊂ S is given by P(A). Then, the probability measure obeys the following axioms:

1. Non Negative Axiom: P(A) ≥ 0 ∀ A
2. Normalisation: P(S) = 1,
3. Finite Additivity: If {A1,A2,…Aj,…} is a sequence of mutually exclusive events such that Ai ∩ Aj = ∅ for all i, j, then P(A1 ∪ A2 ∪···∪Aj ∪···) = P(A1)+P(A2)+···+P(Aj)+···.

The axioms are supplemented by two definitions:

1. The conditional probability of A given B is defined by

2. The events A, B are said to be statistically independent if

In the R programming language, axiomatic probability itself is not a particular idea or function. The preceding response explained axiomatic probability mathematically. It gives a collection of axioms—basic ideas—upon which probability theory may be constructed. Non-negativity, normalisation, and additivity are among these axioms, which was previously mentioned.

Working with probability distributions and performing probabilistic computations in R need the application of axiomatic probability concepts. Numerous functions and packages are available in R for probability and statistics, including distributions for specifying probability distributions, p functions for computing cumulative probabilities, and q functions for quantiles, among others.

Axiomatic probability, also known as the measure-theoretic or Kolmogorov’s probability, is a foundational approach to probability theory that establishes a rigorous mathematical framework for understanding random events and uncertainty. It was developed by the Russian mathematician Andrey Kolmogorov in the 1930s and has become the standard framework for modern probability theory.