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sinh x is half the difference of e x and e −x cosh x is the average of e x and e −x In terms of the exponential function : [ 1 ] [ 4 ] Hyperbolic sine: the odd part of the exponential function, that is, sinh x = e x − e − x 2 = e 2 x − 1 2 e x = 1 − e − 2 x 2 e − x . {\displaystyle \sinh x={\frac {e^{x}-e^{-x}}{2}}={\frac {e ...
The easiest way to find the Hall divisors is to write the prime power factorization of the number in question and take any subset of the factors. For example, to find the Hall divisors of 60, its prime power factorization is 2 2 × 3 × 5, so one takes any product of 3, 2 2 = 4, and 5.
Gaussian distribution: probability of a value being more than 6 standard deviations from the mean on a specific side [8] 10 −9: Nano-(n) 1×10 −9: One in 1,000,000,000 3.9×10 −9: Probability of an entry winning the jackpot in the Mega Millions multi-state lottery in the United States* [9] 5.707×10 −9
A random variable follows a hyperbolic secant distribution if its probability density function can be related to the following standard form of density function by a location and shift transformation: = , where "sech" denotes the hyperbolic secant function.
Since cosh x + sinh x = e x, an analog to de Moivre's formula also applies to the hyperbolic trigonometry. For all integers n, ( + ) = + . If n is a rational number (but not necessarily an integer), then cosh nx + sinh nx will be one of the values of (cosh x + sinh x) n. [4]
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 ...
For all inverse hyperbolic functions, the principal value may be defined in terms of principal values of the square root and the logarithm function. However, in some cases, the formulas of § Definitions in terms of logarithms do not give a correct principal value, as giving a domain of definition which is too small and, in one case non-connected.
If the rolling curve is a circle and the fixed curve is a line then the roulette is a trochoid. If, in this case, the point lies on the circle then the roulette is a cycloid . A related concept is a glissette , the curve described by a point attached to a given curve as it slides along two (or more) given curves.