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Independence is a fundamental notion in probability theory, as in statistics and the theory of stochastic processes.Two events are independent, statistically independent, or stochastically independent [1] if, informally speaking, the occurrence of one does not affect the probability of occurrence of the other or, equivalently, does not affect the odds.
Given two independent events, if the first event can yield one of n equiprobable outcomes and another has one of m equiprobable outcomes then there are mn equiprobable outcomes of the joint event. This means that if log 2 ( n ) bits are needed to encode the first value and log 2 ( m ) to encode the second, one needs log 2 ( mn ) = log 2 ( m ...
In probability theory, an event is a subset of outcomes of an experiment (a subset of the sample space) to which a probability is assigned. [1] A single outcome may be an element of many different events, [2] and different events in an experiment are usually not equally likely, since they may include very different groups of outcomes. [3]
Each of two urns contains twice as many red balls as blue balls, and no others, and one ball is randomly selected from each urn, with the two draws independent of each other. Let A {\displaystyle A} and B {\displaystyle B} be discrete random variables associated with the outcomes of the draw from the first urn and second urn respectively.
This follows from the definition of independence in probability: the probabilities of two independent events happening, given a model, is the product of the probabilities. This is particularly important when the events are from independent and identically distributed random variables , such as independent observations or sampling with replacement .
In probability theory and statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable.The general form of its probability density function is [2] [3] = ().
In probability theory and statistics, the beta distribution is a family of continuous probability distributions defined on the interval [0, 1] or (0, 1) in terms of two positive parameters, denoted by alpha (α) and beta (β), that appear as exponents of the variable and its complement to 1, respectively, and control the shape of the distribution.
If two independent events are measured separately, the total amount of information is the sum of the self-informations of the individual events. The detailed derivation is below, but it can be shown that there is a unique function of probability that meets these three axioms, up to a multiplicative scaling factor.