A concept in probability theory and measure theory. For the most common and most important case of a product of spaces, see th
be a given distribution (or, more generally, a measure) on $ S _ {i} $,
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...at gives the integral representation of that linear functional. A positive distribution can only exist if $L ( p ) > 0$ for any polynomial $p$ that is positive on
...moments" $m _ { i j } = \langle f _ { i } , f _ { j } \rangle$. Finding a distribution for an integral representation of $L$ on $\mathcal{R}$ is called a strong m
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...with the property that if the observed value of a [[random variable]], the distribution of which is connected with the tested hypothesis, falls in that region, the
be the tested hypothesis concerning the distribution of a random variable $ X $,
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...alized Riemann hypothesis in various classical problems in additive number theory. There are also various other improvements of density theorems.
...n="top"> A.F. Lavrik, "A survey of Linnik's large sieve and the density theory of zeros of L-functions" ''Russian Math. Surveys'' , '''35''' : 2 (1980)
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See also [[Prime number]]; [[Distribution of prime numbers]].
[[Category:Number theory]]
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{{MSC|03E04}} ''in set theory''
{{MSC|28A}} ''in measure theory''
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...90 : ~/encyclopedia/old_files/data/I050/I.0500910 Infinitely\AAhdivisible distribution
[[Category:Distribution theory]]
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''(in probability theory)''
...If the distribution of some pair (say, $\alpha$ and $\beta$) is fixed, the distribution of all other parameters can be uniquely calculated (and is not necessarily
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A result in the theory of fluctuations in random walks (cf. [[Random walk|Random walk]]). Let $
be independent random variables with the same distribution (cf. [[Random variable|Random variable]]), and let $ S _ {0} = 0 $,
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...(cf. also [[Density of a probability distribution|Density of a probability distribution]]) is a scalar function $f _ { X } ( X )$ such that:
The matrix variate normal distribution
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...m of constructing similar tests (cf. [[Similar test|Similar test]]) in the theory of statistical hypothesis testing, and the term "Neyman structure" is use
with distribution in the family $ \{ {P _ {e} ^ {T} } : {e \in \Theta _ {0} } \} $.
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...tion that links a multi-dimensional [[Probability distribution|probability distribution]] function to its one-dimensional margins. Such functions first made their
...e and continuous, and hence a continuous bivariate [[Distribution function|distribution function]] on the unit square, with uniform margins. Furthermore, setting
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The de la Vallée-Poussin theorem on the distribution of prime numbers: Let $\pi(x)$ be the number of primes smaller than $x$; th
...$x$. This theorem demonstrates the correctness of Gauss' hypothesis on the distribution of prime numbers, viz., as $x \rightarrow \infty$,
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''in analytic number theory''
Five conjectures, formulated by B. Riemann (1876), concerning the distribution of the non-trivial zeros of the [[Zeta-function|zeta-function]]
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...some kind of statistical regularity (see [[Probability theory|Probability theory]]). One distinguishes random numbers, being generated by a stochastic appar
with distribution function $ F ( z) $;
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...he kinetic [[Boltzmann equation|Boltzmann equation]] for a single-particle distribution function $ f ( t , r , v ) $,
...ccessive approximations, in which the local [[Maxwell distribution|Maxwell distribution]] $ f _ { \mathop{\rm loc} } ( r , v \mid n , u , \theta ) $,
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has a given polynomial distribution, characterized by a vector of positive probabilities $ p = ( p _ {1} \dot
a [[Chi-squared distribution| "chi-squared" distribution]] with $ k - 1 $
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...has to be clearly stated, even in an elementary exposition on probability theory. In the academic literature this point of view was adopted by W. Feller (se
...bility distribution]] of $X$. This measure is uniquely determined by the [[distribution function]] of $X$:
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..., who pointed out that the development of [[Probability theory|probability theory]] and the extension of its domain of applications had led to the necessity
The basic concepts of probability theory, such as the [[Characteristic function|characteristic function]], the [[Mat
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in a Euclidean space and let the probability distribution of this vector belong to the parametric family of distributions defined by
...this problem and its constructive description form the base of the general theory of statistical hypothesis testing.
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