Sample areas in the new investigations were selected strictly by application of the principles of probability theory, so as to be representative of the total population of defined areas within calculable limits.

This list could be expanded to include most fields of mathematics, including measure theory, ergodic theory, probability, representation theory, and differential geometry.

Occasionally, " almost all " is used in the sense of " almost everywhere " in measure theory, or in the closely related sense of " almost surely " in probability theory.

The concept and theory of Kolmogorov Complexity is based on a crucial theorem first discovered by Ray Solomonoff, who published it in 1960, describing it in " A Preliminary Report on a General Theory of Inductive Inference " as part of his invention of algorithmic probability.

In information theory, one bit is typically defined as the uncertainty of a binary random variable that is 0 or 1 with equal probability, or the information that is gained when the value of such a variable becomes known.

Pascal was an important mathematician, helping create two major new areas of research: he wrote a significant treatise on the subject of projective geometry at the age of sixteen, and later corresponded with Pierre de Fermat on probability theory, strongly influencing the development of modern economics and social science.

In computational complexity theory, BPP, which stands for bounded-error probabilistic polynomial time is the class of decision problems solvable by a probabilistic Turing machine in polynomial time, with an error probability of at most 1 / 3 for all instances.

In computational complexity theory, BQP ( bounded error quantum polynomial time ) is the class of decision problems solvable by a quantum computer in polynomial time, with an error probability of at most 1 / 3 for all instances.

Following the work on expected utility theory of Ramsey and von Neumann, decision-theorists have accounted for rational behavior using a probability distribution for the agent.

Johann Pfanzagl completed the Theory of Games and Economic Behavior by providing an axiomatization of subjective probability and utility, a task left uncompleted by von Neumann and Oskar Morgenstern: their original theory supposed that all the agents had the same probability distribution, as a convenience.

The " Ramsey test " for evaluating probability distributions is implementable in theory, and has kept experimental psychologists occupied for a half century.

Combinatorial problems arise in many areas of pure mathematics, notably in algebra, probability theory, topology, and geometry, and combinatorics also has many applications in optimization, computer science, ergodic theory and statistical physics.

In part, the growth was spurred by new connections and applications to other fields, ranging from algebra to probability, from functional analysis to number theory, etc.

Analytic combinatorics concerns the enumeration of combinatorial structures using tools from complex analysis and probability theory.

In probability theory and statistics, the cumulative distribution function ( CDF ), or just distribution function, describes the probability that a real-valued random variable X with a given probability distribution will be found at a value less than or equal to x.

This is totally spurious, since no matter who measured first the other will measure the opposite spin despite the fact that ( in theory ) the other has a 50 % ' probability ' ( 50: 50 chance ) of measuring the same spin, unless data about the first spin measurement has somehow passed faster than light ( of course TI gets around the light speed limit by having information travel backwards in time instead ).

In the computer science subfield of algorithmic information theory, a Chaitin constant ( Chaitin omega number ) or halting probability is a real number that informally represents the probability that a randomly constructed program will halt.

where the right-hand side represents the probability that the random variable X takes on a value less than or

In this way, Ω < sub > F </ sub > represents the probability that a randomly selected infinite sequence of 0s and 1s begins with a bit string ( of some finite length ) that is in the domain of F. It is for this reason that Ω < sub > F </ sub > is called a halting probability.

First, although home glucose meter readings are often misleading, the probability that a low reading, whether accompanied by symptoms or not, represents real hypoglycemia is much higher in a person who takes insulin than in someone who does not.

These are cross-sections of the probability density that are color-coded ( black represents zero density and white represents the highest density ).

The sum of all resulting arrows represents the total probability of the event.

The probability of occurrence is likewise commonly assessed on a scale from 1 to 5, where 1 represents a very low probability of the risk event actually occurring while 5 represents a very high probability of occurrence.

The modern formulation of statistical mechanics is based on the description of the physical system by an ensemble that represents all possible configurations of the system and the probability of realizing each configuration.

In this case X < sub > n </ sub > = ( 1 − ( 1 / 50 )< sup > 6 </ sup >)< sup > n </ sup > where X < sub > n </ sub > represents the probability that none of the first n monkeys types banana correctly on their first try.

In probability theory, the sigma algebra often represents the set of available information, and a function ( in this context a random variable ) is measurable if and only if it represents an outcome that is knowable based on the available information.

Whether it fits significantly better and should thus be preferred is determined by deriving the probability or p-value of the difference D. Where the null hypothesis represents a special case of the alternative hypothesis, the probability distribution of the test statistic is approximately a chi-squared distribution with degrees of freedom equal to df2 − df1.

where the factor represents the impact of on the probability of.

In this expression, W < sub > c, v </ sub >( E ) represents the product of the Brillouin zone-averaged transition probability at the energy E with the joint density of states, J < sub > c, v </ sub >( E ); φ is a broadening function, representing the role of scattering in smearing out the energy levels.

An exclusive OR gate with two inputs represents the probability that one or the other input, but not both, occurs:

To see the elongated shape of ψ ( x, y, z )< sup > 2 </ sup > functions that show probability density more directly, see the graphs of d-orbitals below.

It has the probability density function

The Cauchy distribution has the probability density function

1 is called the standard Cauchy distribution with the probability density function

which is just the Fourier transform of the probability density.

The original probability density may be expressed in terms of the characteristic function, essentially by using the inverse Fourier transform:

If a probability distribution has a density function f ( x ), then the mean is

Together with particle density and path length, it can be used to predict the total scattering probability via the Beer-Lambert law.

** Normal dynamics, is a stochastic motion having a Gaussian probability density function in position with variance MSD that follows, MSD ~ t, where MSD is the mean squared displacement of the process, and t is the time the process is seen ( normal dynamics and Brownian dynamics are very similar ; the term used depends on the field )

Firstly, in estimating the probability density functions of random variables and secondly in estimating the spectral density function of a time series.

Histograms are used to plot density of data, and often for density estimation: estimating the probability density function of the underlying variable.

The total area of a histogram used for probability density is always normalized to 1.

This will construct a smooth probability density function, which will in general more accurately reflect the underlying variable.

The probability density in three-dimensional space is obtained by rotating the one shown here around the z-axis.

Black lines occur in each but the first orbital: these are the nodes of the wavefunction, i. e. where the probability density is zero.

Then the probability density function f *( x ) of the size biased population is

Formally, this means that the probability density functions or probability mass functions in this class have the form

In this case, the probability density function or probability mass function will be a special case of the more general form