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The categorical distribution is the generalization of the Bernoulli distribution for variables with any constant number of discrete values. The Beta distribution is the conjugate prior of the Bernoulli distribution. [5] The geometric distribution models the number of independent and identical Bernoulli trials needed to get one success.
A Bernoulli process is a finite or infinite sequence of independent random variables X 1, X 2, X 3, ..., such that for each i, the value of X i is either 0 or 1; for all values of , the probability p that X i = 1 is the same. In other words, a Bernoulli process is a sequence of independent identically distributed Bernoulli trials.
Graphs of probability P of not observing independent events each of probability p after n Bernoulli trials vs np for various p.Three examples are shown: Blue curve: Throwing a 6-sided die 6 times gives a 33.5% chance that 6 (or any other given number) never turns up; it can be observed that as n increases, the probability of a 1/n-chance event never appearing after n tries rapidly converges to 0.
Nicolaus Bernoulli described the St. Petersburg paradox (involving infinite expected values) in 1713, prompting two Swiss mathematicians to develop expected utility theory as a solution. Bernoulli's paper was the first formalization of marginal utility, which has broad application in economics in addition to expected utility theory. He used ...
Bernoulli was very proud of this result, referring to it as his "golden theorem", [25] and remarked that it was "a problem in which I've engaged myself for twenty years". [26] This early version of the law is known today as either Bernoulli's theorem or the weak law of large numbers, as it is less rigorous and general than the modern version. [27]
In these cases, standard techniques for solving equations of those forms can be applied. For n ≠ 0 {\displaystyle n\neq 0} and n ≠ 1 {\displaystyle n\neq 1} , the substitution u = y 1 − n {\displaystyle u=y^{1-n}} reduces any Bernoulli equation to a linear differential equation
The continuous Bernoulli can be thought of as a continuous relaxation of the Bernoulli distribution, which is defined on the discrete set {,} by the probability mass function: = (), where is a scalar parameter between 0 and 1.
In mathematics, the Bernoulli scheme or Bernoulli shift is a generalization of the Bernoulli process to more than two possible outcomes. [ 1 ] [ 2 ] Bernoulli schemes appear naturally in symbolic dynamics , and are thus important in the study of dynamical systems .