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The law of total probability is [1] a theorem that states, in its discrete case, if is a finite or countably infinite set of mutually exclusive and collectively exhaustive events, then for any event. or, alternatively, [1] where, for any , if , then these terms are simply omitted from the summation since is finite.
Bayes' theorem (alternatively Bayes' law or Bayes' rule, after Thomas Bayes) gives a mathematical rule for inverting conditional probabilities, allowing us to find the probability of a cause given its effect. [1] For example, if the risk of developing health problems is known to increase with age, Bayes' theorem allows the risk to an individual ...
Law of total expectation. The proposition in probability theory known as the law of total expectation, [1] the law of iterated expectations[2] (LIE), Adam's law, [3] the tower rule, [4] and the smoothing theorem, [5] among other names, states that if is a random variable whose expected value is defined, and is any random variable on the same ...
Law of total variance. In probability theory, the law of total variance[1] or variance decomposition formula or conditional variance formulas or law of iterated variances also known as Eve's law, [2] states that if and are random variables on the same probability space, and the variance of is finite, then. In language perhaps better known to ...
This theorem makes rigorous the intuitive notion of probability as the expected long-run relative frequency of an event's occurrence. It is a special case of any of several more general laws of large numbers in probability theory. Chebyshev's inequality. Let X be a random variable with finite expected value μ and finite non-zero variance σ 2.
The standard probability axioms are the foundations of probability theory introduced by Russian mathematician Andrey Kolmogorov in 1933. [1] These axioms remain central and have direct contributions to mathematics, the physical sciences, and real-world probability cases. [2]
In probability theory and statistics, the law of the unconscious statistician, or LOTUS, is a theorem which expresses the expected value of a function g(X) of a random variable X in terms of g and the probability distribution of X. The form of the law depends on the type of random variable X in question. If the distribution of X is discrete and ...
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.
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