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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.
In other words, the terms random sample and IID are synonymous. In statistics, "random sample" is the typical terminology, but in probability, it is more common to say "IID." Identically distributed means that there are no overall trends — the distribution does not fluctuate and all items in the sample are taken from the same probability ...
Pairwise independence does not imply mutual independence, as shown by the following example attributed to S. Bernstein. [3] Suppose X and Y are two independent tosses of a fair coin, where we designate 1 for heads and 0 for tails.
A set of rules governing statements of conditional independence have been derived from the basic definition. [4] [5] These rules were termed "Graphoid Axioms" by Pearl and Paz, [6] because they hold in graphs, where is interpreted to mean: "All paths from X to A are intercepted by the set B". [7]
In statistics, probability theory and information theory, pointwise mutual information (PMI), [1] or point mutual information, is a measure of association.It compares the probability of two events occurring together to what this probability would be if the events were independent.
The probability is sometimes written to distinguish it from other functions and measure P to avoid having to define "P is a probability" and () is short for ({: ()}), where is the event space, is a random variable that is a function of (i.e., it depends upon ), and is some outcome of interest within the domain specified by (say, a particular ...
The certainty that is adopted can be described in terms of a numerical measure, and this number, between 0 and 1 (where 0 indicates impossibility and 1 indicates certainty) is called the probability. Probability theory is used extensively in statistics , mathematics , science and philosophy to draw conclusions about the likelihood of potential ...
Probability of each word in a sequence is independent from probabilities of other word in the sequence. Each word's probability in the sequence is equal to the word's probability in an entire document. = () (). The model consists of units, each treated as one-state finite automata. [3] Words with their probabilities in a document can be ...