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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 general, uncorrelatedness is not the same as orthogonality, except in the special case where at least one of the two random variables has an expected value of 0. In this case, the covariance is the expectation of the product, and X {\displaystyle X} and Y {\displaystyle Y} are uncorrelated if and only if E [ X Y ] = 0 {\displaystyle ...
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.
(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]
Students of statistics and probability theory sometimes develop misconceptions about the normal distribution, ideas that may seem plausible but are mathematically untrue. For example, it is sometimes mistakenly thought that two linearly uncorrelated, normally distributed random variables must be statistically independent.
Kolmogorov's zero–one law asserts that, if the F n are stochastically independent, then for any event (()), one has either P(E) = 0 or P(E)=1. The statement of the law in terms of random variables is obtained from the latter by taking each F n to be the σ-algebra generated by the random variable X n .
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 ...
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.