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In probability theory and statistics, the Bernoulli distribution, named after Swiss mathematician Jacob Bernoulli, [1] is the discrete probability distribution of a random variable which takes the value 1 with probability and the value 0 with probability =.
Pierre Rémond de Montmort, in collaboration with Nicolaus Bernoulli, wrote a book on probability Essay d'analyse sur les jeux de hazard which appeared in 1708, which can be seen as an extension of the Part III of Ars Conjectandi which applies combinatorics and probability to analyze games of chance commonly played at that time. [34]
Jacob Bernoulli's first important contributions were a pamphlet on the parallels of logic and algebra published in 1685, work on probability in 1685 and geometry in 1687. His geometry result gave a construction to divide any triangle into four equal parts with two perpendicular lines.
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.
In probability and statistics, an urn problem is an idealized mental exercise in which some objects of real interest (such as atoms, people, cars, etc.) are represented as colored balls in an urn or other container. One pretends to remove one or more balls from the urn; the goal is to determine the probability of drawing one color or another ...
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.
The probability density function (PDF) for the Wilson score interval, plus PDF s at interval bounds. Tail areas are equal. Since the interval is derived by solving from the normal approximation to the binomial, the Wilson score interval ( , + ) has the property of being guaranteed to obtain the same result as the equivalent z-test or chi-squared test.
In probability theory, statistics, and machine learning, the continuous Bernoulli distribution [1] [2] [3] is a family of continuous probability distributions parameterized by a single shape parameter (,), defined on the unit interval [,], by: