probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...

and

statistics Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...

, smoothness of a

density function In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can ...

is a measure which determines how many times the density function can be differentiated, or equivalently the limiting behavior of distribution’s characteristic function. Formally, we call the distribution of a

random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...

''X'' ordinary smooth of order ''β'' if its characteristic function satisfies :

d_0 , t, ^ \leq , \varphi_X(t),  \leq d_1 , t, ^ \quad \text t\to\infty

for some positive constants ''d''₀, ''d''₁, ''β''. The examples of such distributions are

gamma Gamma (uppercase , lowercase ; ''gámma'') is the third letter of the Greek alphabet. In the system of Greek numerals it has a value of 3. In Ancient Greek, the letter gamma represented a voiced velar stop . In Modern Greek, this letter re ...

, exponential,

uniform A uniform is a variety of clothing worn by members of an organization while participating in that organization's activity. Modern uniforms are most often worn by armed forces and paramilitary organizations such as police, emergency services, se ...

, etc. The distribution is called supersmooth of order ''β'' if its characteristic function satisfies :

d_0 , t, ^\exp\big(-, t, ^\beta/\gamma\big) \leq , \varphi_X(t),  \leq d_1 , t, ^\exp\big(-, t, ^\beta/\gamma\big) \quad \text t\to\infty

for some positive constants ''d''₀, ''d''₁, ''β'', ''γ'' and constants ''β''₀, ''β''₁. Such supersmooth distributions have derivatives of all orders. Examples: normal, Cauchy, mixture normal.

References

* Theory of probability distributions {{probability-stub