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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.
(That is, the two dice are independent.) If, however, the 1st die's result is a 3, and someone tells you about a third event - that the sum of the two results is even - then this extra unit of information restricts the options for the 2nd result to an odd number. In other words, two events can be independent, but NOT conditionally independent. [2]
Here, p,q are arbitrary (sufficiently smooth) functions of two variables, so (due their modest time dependence) the integrals P,Q also count as "freely chosen" functions of two variables; as promised, one of them is differentiated once before adding to the other to express the general solution of the initial value problem for the two ...
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.
Total correlation quantifies the amount of dependence among a group of variables. A near-zero total correlation indicates that the variables in the group are essentially statistically independent; they are completely unrelated, in the sense that knowing the value of one variable does not provide any clue as to the values of the other variables.
In probability theory and information theory, the mutual information (MI) of two random variables is a measure of the mutual dependence between the two variables. More specifically, it quantifies the "amount of information" (in units such as shannons , nats or hartleys) obtained about one random variable by observing the other random variable.
More generally, one can refer to the conditional distribution of a subset of a set of more than two variables; this conditional distribution is contingent on the values of all the remaining variables, and if more than one variable is included in the subset then this conditional distribution is the conditional joint distribution of the included ...
Independent: Each outcome of the die roll will not affect the next one, which means the 10 variables are independent from each other. Identically distributed: Regardless of whether the die is fair or weighted, each roll will have the same probability of seeing each result as every other roll. In contrast, rolling 10 different dice, some of ...