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The Bernoulli distribution is a special case of the binomial distribution with = [4] The kurtosis goes to infinity for high and low values of p , {\displaystyle p,} but for p = 1 / 2 {\displaystyle p=1/2} the two-point distributions including the Bernoulli distribution have a lower excess kurtosis , namely −2, than any other probability ...
Before Daniel Bernoulli's 1738 publication, mathematician Gabriel Cramer from Geneva had already in 1728 found parts of this idea (also motivated by the St. Petersburg paradox), stating that the mathematicians estimate money in proportion to its quantity, and men of good sense in proportion to the usage that they may make of it.
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 moments of a function are certain quantitative measures related to the shape of the function's graph.If the function represents mass density, then the zeroth moment is the total mass, the first moment (normalized by total mass) is the center of mass, and the second moment is the moment of inertia.
This random variable X has a Bernoulli distribution with parameter . [29] This is a transformation of discrete random variable. For a distribution function F {\displaystyle F} of an absolutely continuous random variable, an absolutely continuous random variable must be constructed.
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 ...
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