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In mathematics, the multiplication theorem is a certain type of identity obeyed by many special functions related to the gamma function. For the explicit case of the gamma function, the identity is a product of values; thus the name.
This theorem follows from the fact that if X n converges in distribution to X and Y n converges in probability to a constant c, then the joint vector (X n, Y n) converges in distribution to (X, c) . Next we apply the continuous mapping theorem , recognizing the functions g ( x , y ) = x + y , g ( x , y ) = xy , and g ( x , y ) = x y −1 are ...
The normal number theorem (1909), due to Émile Borel, could be one of the first examples of the probabilistic method, providing the first proof of existence of normal numbers, with the help of the first version of the strong law of large numbers (see also the first item of the section Analysis).
It should only contain pages that are Probability theorems or lists of Probability theorems, as well as subcategories containing those things (themselves set categories). Topics about Probability theorems in general should be placed in relevant topic categories .
A diagram representing a two-state Markov chain. The states are represented by 'A' and 'E'. The numbers are the probability of flipping the state. Discrete mathematics, broadly speaking, is the study of individual, countable mathematical objects. An example is the set of all integers. [42]
Methods of probability theory also apply to descriptions of complex systems given only partial knowledge of their state, as in statistical mechanics or sequential estimation. A great discovery of twentieth-century physics was the probabilistic nature of physical phenomena at atomic scales, described in quantum mechanics .