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In probability theory, the chain rule [1] (also called the general product rule [2] [3]) describes how to calculate the probability of the intersection of, not necessarily independent, events or the joint distribution of random variables respectively, using conditional probabilities.
The rule for integration by parts is derived from the product rule, as is (a weak version of) the quotient rule. (It is a "weak" version in that it does not prove that the quotient is differentiable but only says what its derivative is if it is differentiable.)
A product distribution is a probability distribution constructed as the distribution of the product of random variables having two other known distributions. Given two statistically independent random variables X and Y, the distribution of the random variable Z that is formed as the product = is a product distribution.
In combinatorics, the rule of product or multiplication principle is a basic counting principle (a.k.a. the fundamental principle of counting). Stated simply, it is the intuitive idea that if there are a ways of doing something and b ways of doing another thing, then there are a · b ways of performing both actions. [1] [2]
The product of the prior and the likelihood, when normalized, results in a posterior probability distribution that incorporates all the information known to date. [9] By Aumann's agreement theorem , Bayesian agents whose prior beliefs are similar will end up with similar posterior beliefs.
the product of two random variables is a random variable; addition and multiplication of random variables are both commutative ; and there is a notion of conjugation of random variables, satisfying ( XY ) * = Y * X * and X ** = X for all random variables X , Y and coinciding with complex conjugation if X is a constant.
The rule of sum is an intuitive principle stating that if there are a possible outcomes for an event (or ways to do something) and b possible outcomes for another event (or ways to do another thing), and the two events cannot both occur (or the two things can't both be done), then there are a + b total possible outcomes for the events (or total possible ways to do one of the things).
When X n converges in r-th mean to X for r = 1, we say that X n converges in mean to X. When X n converges in r-th mean to X for r = 2, we say that X n converges in mean square (or in quadratic mean) to X. Convergence in the r-th mean, for r ≥ 1, implies convergence in probability (by Markov's inequality).
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