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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.
Bernoulli's principle can be used to calculate the lift force on an airfoil, if the behaviour of the fluid flow in the vicinity of the foil is known. For example, if the air flowing past the top surface of an aircraft wing is moving faster than the air flowing past the bottom surface, then Bernoulli's principle implies that the pressure on the ...
For the Bernoulli polynomials, the number of crossings of the x-axis in the unit interval does not go up with the degree. In the limit of large degree, they approach, when appropriately scaled, the sine and cosine functions. A similar set of polynomials, based on a generating function, is the family of Euler polynomials.
The Bernoulli equation applicable to incompressible flow shows that the stagnation pressure is equal to the dynamic pressure and static pressure combined. [ 1 ] : § 3.5 In compressible flows , stagnation pressure is also equal to total pressure as well, provided that the fluid entering the stagnation point is brought to rest isentropically .
A Binomial distributed random variable X ~ B(n, p) can be considered as the sum of n Bernoulli distributed random variables. So the sum of two Binomial distributed random variables X ~ B(n, p) and Y ~ B(m, p) is equivalent to the sum of n + m Bernoulli distributed random variables, which means Z = X + Y ~ B(n + m, p). This can also be proven ...
The remainder term has an exact expression in terms of the periodized Bernoulli functions P k (x). The Bernoulli polynomials may be defined recursively by B 0 (x) = 1 and, for k ≥ 1, ′ = (), = The periodized Bernoulli functions are defined as = (⌊ ⌋), where ⌊x⌋ denotes the largest integer less than or equal to x, so that x − ⌊x ...
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 probability theory, a Chernoff bound is an exponentially decreasing upper bound on the tail of a random variable based on its moment generating function.The minimum of all such exponential bounds forms the Chernoff or Chernoff-Cramér bound, which may decay faster than exponential (e.g. sub-Gaussian).