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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 ...
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 probability space, then.
The rule is often called Chebyshev's theorem, about the range of standard deviations around the mean, in statistics. The inequality has great utility because it can be applied to any probability distribution in which the mean and variance are defined. For example, it can be used to prove the weak law of large numbers.
Probability theory. In probability theory, the law (or formula) of total probability is a fundamental rule relating marginal probabilities to conditional probabilities. It expresses the total probability of an outcome which can be realized via several distinct events, hence the name.
Probability theory. 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]
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.
The law of large numbers provides an expectation of an unknown distribution from a realization of the sequence, but also any feature of the probability distribution.[1] By applying Borel's law of large numbers, one could easily obtain the probability mass function.
Cochran's theorem then states that Q1 and Q2 are independent, with chi-squared distributions with n − 1 and 1 degree of freedom respectively. This shows that the sample mean and sample variance are independent. This can also be shown by Basu's theorem, and in fact this property characterizes the normal distribution – for no other ...