For an example of its use, analysis of the concentration of elements is important in managing a nuclear reactor, so nuclear scientists will analyze neutron activation to develop discrete measurements within vast samples.

For example, an 8-bit CPU deals with a range of numbers that can be represented by eight binary digits ( each digit having two possible values ), that is, 2 < sup > 8 </ sup > or 256 discrete numbers.

For the study of the mechanical behavior of solids and fluids these are assumed to be continuous bodies, which means that the matter fills the entire region of space it occupies, despite the fact that matter is made of atoms, has voids, and is discrete.

For example, the real line equipped with the discrete topology is closed and bounded but not compact, as the collection of all singleton points of the space is an open cover which admits no finite subcover.

For microscopic, atomic-level systems like a molecule, angular momentum can only have specific discrete values given by

For example, when a laboratory apparatus was developed that could reliably fire one electron at a time through the double slit, the emergence of an interference pattern suggested that each electron was interfering with itself, and therefore in some sense the electron had to be going through both slits at once — an idea that contradicts our everyday experience of discrete objects.

For real inputs, the real part of is none other than the discrete Hartley transform, which is also involutary.

For elliptic-curve-based protocols, it is assumed that finding the discrete logarithm of a random elliptic curve element with respect to a publicly known base point is infeasible.

For more than 40 years, Paul Ekman has supported the view that emotions are discrete, measurable, and physiologically distinct.

For example, JPEG compression uses a variant of the Fourier transformation ( discrete cosine transform ) of small square pieces of a digital image.

For periodic functions, both the Fourier transform and the DTFT comprise only a discrete set of frequency components ( Fourier series ), and the transforms diverge at those frequencies.

For a population of discrete values, or for a continuous population density, the th-quantile is the data value where the cumulative distribution function crosses.

( For comparison, a terabyte of digital information stores only 2 < sup > 43 </ sup > discrete on / off values.

For instance, it is known from quantum mechanics that certain aspects of electromagnetism involve discrete particles — photons — rather than continuous fields.

For n independent and identically distributed discrete random variables X < sub > 1 </ sub >, X < sub > 2 </ sub >, ..., X < sub > n </ sub > with cumulative distribution function G ( x ) and probability mass function g ( x ) the range of the X < sub > i </ sub > is the range of a sample of size n from a population with distribution function G ( x ).

For Leibniz, then, space was an idealised abstraction from the relations between individual entities or their possible locations and therefore could not be continuous but must be discrete.

For example, if J is a discrete category, the components of the unit are the diagonal morphisms δ: N → N < sup > J </ sup >.

For discrete devices, for example, there are three standards: JEDEC JESD370B in United States, Pro Electron in Europe and Japanese Industrial Standards ( JIS ) in Japan.

For example, the product of the unit circle ( with its usual topology ) and the real line with the discrete topology is a locally compact group with the product topology and Haar measure on this group is not inner regular for the closed subset

For the definition of the likelihood function, one has to distinguish between discrete and continuous probability distributions.

For an observation from the discrete component, the probability can either be written down directly or treated within the above context by saying that the probability of getting an observation in an interval that does contain a discrete component ( of being in interval j which contains discrete component k ) is approximately

For example, the abstract concept of number springs from the experience of counting discrete objects.

For him, " probability " means a higher chance of occurring, and brings about a higher degree of subjective expectation in the viewer.

* For the frequentist a hypothesis is a proposition ( which must be either true or false ), so that the frequentist probability of a hypothesis is either one or zero.

For objectivists, probability objectively measures the plausibility of propositions, i. e. the probability of a proposition corresponds to a reasonable belief everyone ( even a " robot ") sharing the same knowledge should share in accordance with the rules of Bayesian statistics, which can be justified by requirements of rationality and consistency.

For subjectivists probability corresponds to a ' personal belief '.

For example, Hacking writes " And neither the Dutch book argument, nor any other in the personalist arsenal of proofs of the probability axioms, entails the dynamic assumption.

For example, the probability that an alpha particle striking a beryllium target will produce a neutron can be expressed as the equivalent cross section of beryllium for this type of reaction.

For example, in a system where there is no queuing, the GoS may be that no more than 1 call in 100 is blocked ( i. e., rejected ) due to all circuits being in use ( a GoS of 0. 01 ), which becomes the target probability of call blocking, P < sub > b </ sub >, when using the Erlang B formula.

For example, if one flips a fair coin 21 times, then the probability of 21 heads is 1 in 2, 097, 152 ( above ).

For most types of hashing functions the choice of the function depends strongly on the nature of the input data, and their probability distribution in the intended application.

For any information rate R < C and coding error ε > 0, for large enough N, there exists a code of length N and rate ≥ R and a decoding algorithm, such that the maximal probability of block error is ≤ ε ; that is, it is always possible to transmit with arbitrarily small block error.

For instance, if a particle is in a state | ψ ⟩, the probability of finding it in a region of volume d < sup > 3 </ sup > x surrounding some position x is

For example, consider a model which gives the probability density function of observable random variable X as a function of a parameter θ.

For example, the result of a significance test depends on the probability of a result as extreme or more extreme than the observation, and that probability may depend on the design of the experiment.

For a probability density function f ( x ) with the cumulative distribution function F ( x ), the Lorenz curve L ( F ( x )) is given by:

For a probability distribution, the mean is equal to the sum or integral over every possible value weighted by the probability of that value.

For example, naive Bayes and linear discriminant analysis are joint probability models, whereas logistic regression is a conditional probability model.

For the special case where is a joint probability distribution and the loss function is the negative log likelihood a risk minimization algorithm is said to perform generative training, because can be regarded as a generative model that explains how the data were generated.

For calling to have a positive expectation, Alice must believe the probability of her opponent having a weak hand is over 40 percent.

For example, seven shuffles of a new deck leaves an 81 % probability of winning New Age Solitaire where the probability is 50 % with a uniform random deck ( Mann, especially section 10 ).