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More formally, a multivariate random variable is a column vector X = ( X < sub > 1 </ sub >, ..., X < sub > n </ sub >)< sup > T </ sup > ( or its transpose, which is a row vector ) whose components are scalar-valued random variables on the same probability space ( Ω,, P ), where Ω is the sample space, is the sigma-algebra ( the collection of all events ), and P is the probability measure ( a function returning every event's probability ).
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More and formally
More formally, it is the theoretical temperature at which entropy reaches its minimum value.
More formally, the complexity of a string is the length of the shortest possible description of the string in some fixed universal description language ( the sensitivity of complexity relative to the choice of description language is discussed below ).
More formally, a block cipher is specified by an encryption function
More formally a k-combination of a set S is a subset of k distinct elements of S. If the set has n elements the number of k-combinations is equal to the binomial coefficient
More formally, discrete mathematics has been characterized as the branch of mathematics dealing with countable sets ( sets that have the same cardinality as subsets of the natural numbers, including rational numbers but not real numbers ).
More formally design has been defined as follows.
More formally, it is " the study of mathematical models of conflict and cooperation between intelligent rational decision-makers.
More formally, such coordinates can be obtained from three-dimensional coordinates using the artifice of a map projection.
More formally, a " representation " means a homomorphism from the group to the automorphism group of an object.
More formally, case has been defined as " a system of marking dependent nouns for the type of relationship they bear to their heads.
More formally, if a, b, c, and d are all integers, then the fraction < sup > a </ sup >⁄< sub > b </ sub > is irreducible if and only if there is no other equal fraction < sup > c </ sup >⁄< sub > d </ sub > such that | c | < | a | or | d | < | b |, where | a | means the absolute value of a.
More formally, the Klein bottle is the quotient space described as the square × with sides identified by the relations for and for:
More formally, a lexicon is a language's inventory of lexemes.
More formally, the environment is modeled as a Markov decision process ( MDP ) with states and actions with the following probability distributions: the instantaneous cost distribution, the observation distribution and the transition, while a policy is defined as conditional distribution over actions given the observations.
( More work is required to derive this result formally.
More formally, statistical theory defines a statistic as a function of a sample where the function itself is independent of the sample's distribution ; that is, the function can be stated before realization of the data.
More formally, # P is the class of function problems of the form " compute ƒ ( x )," where ƒ is the number of accepting paths of a nondeterministic Turing machine running in polynomial time.
More formally, the output must satisfy two conditions:
More formally, we can state the Transfinite Recursion Theorem as follows.
More formally, ubiquitous computing is defined as " machines that fit the human environment instead of forcing humans to enter theirs.
More formally, complex systems often exhibit hysteresis.
More formally, two sets of points are called congruent if, and only if, one can be transformed into the other by an isometry, i. e., a combination of translations, rotations and reflections.
More formally, linear programming is a technique for the optimization of a linear objective function, subject to linear equality and linear inequality constraints.
More formally, ( X, Σ, μ ) is complete if and only if
More formally, we can think of a truth condition as what makes for the truth of a sentence in an inductive definition of truth ( for details, see the semantic theory of truth ).
More and multivariate
More complex multivariate analyses are sometimes used when multiple characteristics are involved and a univariate analysis could produce confounded results.
More advanced statistical methods employed by some I – O psychologists include logistic regression, multivariate analysis of variance, structural equation modeling, and hierarchical linear modeling ( HLM ; also known as multilevel modeling ).
More generally, the multivariate resultant or Macaulay's resultant of n homogeneous polynomials in n variables is a polynomial in their coefficients that vanishes when they have a common non-zero solution, or equivalently when the n hypersurfaces corresponding to them have a common zero in n – 1 dimensional projective space.
More and random
More precisely, he showed that a random graph on vertices, formed by choosing independently whether to include each edge with probability has, with probability tending to 1 as goes to infinity, at most cycles of length or less, but has no independent set of size Therefore, removing one vertex from each short cycle leaves a smaller graph with girth greater than in which each color class of a coloring must be small and which therefore requires at least colors in any coloring.
More generally, if X < sub > 1 </ sub >, ..., X < sub > n </ sub > are independent random variables, not necessarily identically distributed, but all having the same variance, then
More specifically, Bob's qubit b is in one of four states uniformly chosen at random and Bob cannot obtain any information about c from his qubit.
More substantially, the terms statistical inference, statistical induction and inferential statistics are used to describe systems of procedures that can be used to draw conclusions from datasets arising from systems affected by random variation, such as observational errors, random sampling, or random experimentation.
More generally, data about a random process is obtained from its observed behavior during a finite period of time.
More generally, semi-parametric models can often be separated into ' structural ' and ' random variation ' components.
More evidence for this being the case is the way that good human players find it much easier to recall positions from genuine chess games, breaking them down into a small number of recognizable sub-positions, rather than completely random arrangements of the same pieces.
More generally, random selection of decision makers from a larger group is known as sortition ( from the Latin base for lottery ).
More experimentally, stereo or multichannel scattering, random reordering, disintegration and morphing are possible.
More precisely, suppose X < sub > 1 </ sub >, X < sub > 2 </ sub >, X < sub > 3 </ sub >, ... is an infinite exchangeable sequence of Bernoulli-distributed random variables.
More recently ( 2001 and 2002 ), Bailey and Richard Crandall showed that the existence of this and similar formulas has implications for the long-standing question of " normality " – whether and why the digits of certain mathematical constants ( including pi ) appear " random " in a particular sense.
" More quantitatively, it is one in which the standard deviation of average annual returns declines faster than the inverse of the holding period, implying that the process is not a random walk, but that periods of lower returns are systematically followed by compensating periods of higher returns.
More generally, where < sup > T </ sup >, an n-dimensional random vector, one uses instead of tX:
More generally, let T be a real topological vector space, and X a T-valued integrable random variable.
* More generally, for every integrable N – measurable random variable Y on Ω.
More specifically, a Turing machine can be simulated with arbitrarily high probability of running correctly for all time, only if a random CRN is used.
More generally, the values might be defined over a continuous domain, and the random field might be thought of as a " function valued " random variable.
More precisely, scaling in stochastic systems concerns itself with the likelihood of choosing a particular configuration out of the set of all possible random configurations.
More technically, MPT models an asset's return as a normally distributed function ( or more generally as an elliptically distributed random variable ), defines risk as the standard deviation of return, and models a portfolio as a weighted combination of assets, so that the return of a portfolio is the weighted combination of the assets ' returns.
More abstractly: If X < sub > 1 </ sub >, ..., X < sub > n + 1 </ sub > are conditionally independent random variables that each can assume the value 0 or 1, then, if we know nothing more about them,
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