the term "characteristic function" can refer to any of several distinct concepts: The indicator function of a subset, that is the function 1 A : X → {...

In probability theory and statistics, the characteristic function of any real-valued random variable completely defines its probability distribution....

In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all...

cases. The moment-generating function of a real-valued distribution does not always exist, unlike the characteristic function. There are relations between...

nth derivative of the characteristic function evaluated at t = 0 {\displaystyle t=0} . Observe that the characteristic function is not differentiable...

random variable ⁠ X {\displaystyle X} ⁠ is closely connected to the characteristic function φ X ( t ) {\textstyle \varphi _{X}(t)} of that variable, which...

current–voltage characteristic, the current in a circuit as a function of the applied voltage Receiver operating characteristic Characteristic (algebra) of...

time-independent function W ( q ) {\displaystyle W(\mathbf {q} )} is sometimes called the abbreviated action or Hamilton's characteristic function : 434  and...

restatement: the product of the characteristic functions of a number of density functions becomes close to the characteristic function of the normal density as...

random variables with common characteristic function φ ( t ) {\displaystyle \varphi (t)} . The empirical characteristic function (ECF) defined as φ n ( t...

imaginary part, and hence the characteristic function is not analytic at the origin. Consequently, the characteristic function of the log-normal distribution...

field of mathematics known as convex analysis, the characteristic function of a set is a convex function that indicates the membership (or non-membership)...

\end{aligned}}} The characteristic function is the Fourier transform of the probability density function. The characteristic function of the beta distribution...

_{j=1}^{n}e^{itx_{j}}} Knowing the characteristic function, it is possible to find the corresponding probability density function through the Fourier transform...

probability density function f X {\displaystyle f_{X}} of a scalar random variable X {\displaystyle X} is related to its characteristic function φ X {\displaystyle...

.. have the same characteristic function, so we will simply denote this φX. Among the basic properties of characteristic functions there are φ 1 n X...

important formulas like Paul Lévy's inversion formula for the characteristic function also rely on the "less than or equal" formulation. If treating...

equality, and of Pascal's rule in the second last equality. The characteristic function of each X k {\displaystyle X_{k}} and of Z {\displaystyle Z} is...

respect to the Wigner semicircle distribution of radius 1. The characteristic function of the Wigner distribution can be determined from that of the beta-variate...

cumulant-generating function as the natural logarithm of the characteristic function, which is sometimes also called the second characteristic function, H ( t )...

The characteristic state function or Massieu's potential in statistical mechanics refers to a particular relationship between the partition function of...

Not every function is the characteristic function of a legitimate probability distribution (that is, one whose cumulative distribution function is real...

variance requires uncorrelatedness, but not independence. The characteristic function φ X + Y ( t ) = E ⁡ ( e i t ( X + Y ) ) {\displaystyle \varphi...

its characteristic function (which uniquely determines the distribution) can be acquired by multiplying the corresponding characteristic functions. Consider...

{\displaystyle t=0} we say that the moment generating function does not exist. The characteristic function is given by φ ( t ; α , x m ) = α ( − i x m t ) α...

_{Y}(s).} In particular the characteristic function of their sum is the product of their marginal characteristic functions: φ X + Y ( t ) = φ X ( t ) ⋅...

voltage meter on the right. The temperature Tsense is obtained via a characteristic function E(T) for the type of thermocouple which requires inputs: measured...

players N {\displaystyle N} , called the grand coalition, and a characteristic function v : 2 N → R {\displaystyle v:2^{N}\to \mathbb {R} } from the set...

dependent scalar output (known as a transfer curve or characteristic curve). Transfer functions for components are used to design and analyze systems...

density function (pdf) or probability mass function, the cumulative distribution function (cdf) and the characteristic function. The quantile function, Q,...