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The Poisson–Boltzmann equation describes a model proposed independently by Louis Georges Gouy and David Leonard Chapman in 1910 and 1913, respectively. [3] In the Gouy-Chapman model, a charged solid comes into contact with an ionic solution, creating a layer of surface charges and counter-ions or double layer. [4]
Siméon Denis Poisson. Poisson's equation is an elliptic partial differential equation of broad utility in theoretical physics.For example, the solution to Poisson's equation is the potential field caused by a given electric charge or mass density distribution; with the potential field known, one can then calculate the corresponding electrostatic or gravitational (force) field.
For sufficiently large values of λ, (say λ >1000), the normal distribution with mean λ and variance λ (standard deviation ) is an excellent approximation to the Poisson distribution. If λ is greater than about 10, then the normal distribution is a good approximation if an appropriate continuity correction is performed, i.e., if P( X ≤ x ...
If X is a negative binomial random variable with r large, P near 1, and r(1 − P) = λ, then X approximately has a Poisson distribution with mean λ. Consequences of the CLT: If X is a Poisson random variable with large mean, then for integers j and k, P(j ≤ X ≤ k) approximately equals to P(j − 1/2 ≤ Y ≤ k + 1/2) where Y is a normal ...
The (a,b,0) class of distributions is also known as the Panjer, [1] [2] the Poisson-type or the Katz family of distributions, [3] [4] and may be retrieved through the Conway–Maxwell–Poisson distribution. Only the Poisson, binomial and negative binomial distributions satisfy the full form of this
This is true when the negative Fourier coefficients of f all vanish. In particular, the Poisson kernel is commonly used to demonstrate the equivalence of the Hardy spaces on the unit disk, and the unit circle. The space of functions that are the limits on T of functions in H p (z) may be called H p (T).
In probability theory, the law of rare events or Poisson limit theorem states that the Poisson distribution may be used as an approximation to the binomial distribution, under certain conditions. [1] The theorem was named after Siméon Denis Poisson (1781–1840). A generalization of this theorem is Le Cam's theorem
Without loss of generality, we will take λ to be non-negative. When λ is zero , the equation reduces to Poisson's equation . Therefore, when λ is very small, the solution approaches that of the unscreened Poisson equation, which, in dimension n = 3 {\displaystyle n=3} , is a superposition of 1/ r functions weighted by the source function f :