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In probability theory and statistics, the exponential distribution or negative exponential distribution is the probability distribution of the distance between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate; the distance parameter could be any meaningful mono-dimensional measure of the process, such as time ...
A Gaussian minus exponential distribution has been suggested for modelling option prices. [20] If such a random variable Y has parameters μ , σ , λ , then its negative -Y has an exponentially modified Gaussian distribution with parameters -μ , σ , λ , and thus Y has mean μ − 1 λ {\displaystyle \mu -{\tfrac {1}{\lambda }}} and variance ...
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In probability theory and statistics, the beta distribution is a family of continuous probability distributions defined on the interval [0, 1] or (0, 1) in terms of two positive parameters, denoted by alpha (α) and beta (β), that appear as exponents of the variable and its complement to 1, respectively, and control the shape of the distribution.
In probability and statistics, an exponential family is a parametric set of probability distributions of a certain form, specified below. This special form is chosen for mathematical convenience, including the enabling of the user to calculate expectations, covariances using differentiation based on some useful algebraic properties, as well as for generality, as exponential families are in a ...
In probability theory and statistics, the characteristic function of any real-valued random variable completely defines its probability distribution. If a random variable admits a probability density function, then the characteristic function is the Fourier transform (with sign reversal) of the probability density function.
A quantity is subject to exponential decay if it decreases at a rate proportional to its current value. Symbolically, this process can be expressed by the following differential equation , where N is the quantity and λ ( lambda ) is a positive rate called the exponential decay constant , disintegration constant , [ 1 ] rate constant , [ 2 ] or ...
One of the simplest definitions is: The exponential function is the unique differentiable function that equals its derivative, and takes the value 1 for the value 0 of its variable. This "conceptual" definition requires a uniqueness proof and an existence proof, but it allows an easy derivation of the main properties of the exponential function.