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The earliest solution, however, was offered by Gottfried Leibniz, who published his result in the same year and whose method is the one still used today. [ 5 ] Bernoulli equations are special because they are nonlinear differential equations with known exact solutions.
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 ...
In triangle geometry, the Bernoulli quadrisection problem asks how to divide a given triangle into four equal-area pieces by two perpendicular lines. Its solution by Jacob Bernoulli was published in 1687. [1] [2] Leonhard Euler formulated a complete solution in 1779. [1] [3]
The calculus of variations may be said to begin with Newton's minimal resistance problem in 1687, followed by the brachistochrone curve problem raised by Johann Bernoulli (1696). [2] It immediately occupied the attention of Jacob Bernoulli and the Marquis de l'Hôpital , but Leonhard Euler first elaborated the subject, beginning in 1733.
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.
Can be reduced to a Bernoulli differential equation; a general case of the Jacobi equation [11] Elliptic function: 1 ′ = () Equation for which the elliptic functions are solutions [12] Euler's differential equation: 1
Although Bernoulli deduced that pressure decreases when the flow speed increases, it was Leonhard Euler in 1752 who derived Bernoulli's equation in its usual form. [4] [5] Bernoulli's principle can be derived from the principle of conservation of energy. This states that, in a steady flow, the sum of all forms of energy in a fluid is the same ...
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.