We devote a chapter to the binomial distribution not only because it is a mathematical model for an enormous variety of real life phenomena, but also because it has important properties that recur in many other probability models.

The analysis of variance can be presented in terms of a linear model, which makes the following assumptions about the probability distribution of the responses:

This method, widely used in drug development, is referred to as rNPV ( risk-adjusted NPV ), and similar methods are used to incorporate credit risk in the probability model of CDS valuation.

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

Much like the use of probability in optimal coding theory, rate-distortion theory heavily draws on Bayesian estimation and decision theory in order to model perceptual distortion and even aesthetic judgment.

* Statistical model, in applied statistics, a parameterized set of probability distributions

Although and can be any space of functions, many learning algorithms are probabilistic models where takes the form of a conditional probability model, or takes the form of a joint probability model.

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.

While it is possible to define some arbitrary, ad hoc cost function, frequently a particular cost will be used, either because it has desirable properties ( such as convexity ) or because it arises naturally from a particular formulation of the problem ( e. g., in a probabilistic formulation the posterior probability of the model can be used as an inverse cost ).

Its form depends on the application: for example, in compression it could be related to the mutual information between and, whereas in statistical modeling, it could be related to the posterior probability of the model given the data.

The cache language model and other statistical language models that are used in natural language processing are also examples of applications of probability theory.

This may be obtained from consideration of whether the required prior probability is greater or lesser than a reference probability associated with an urn model or a thought experiment.

In the context of a mathematical model, such as a probability distribution, the distinction between variables and parameters was described by Bard as follows:

Probability clouds are approximate, but better than the Bohr model, whereby electron location is given by a probability function, the wave function eigenvalue, such that the probability is the squared modulus of the complex amplitude, or quantum state nuclear attraction.

Unlike the earlier Bohr model of the atom, however, the wave model describes electrons as " clouds " moving in orbitals, and their positions are represented by probability distributions rather than discrete points.

* Quantum cognition, using quantum probability theory to model human behavior

In mathematical terms, a statistical model is frequently thought of as a pair where is the set of possible observations and the set of possible probability distributions on.

The goal of the divine plan as revealed in Exodus is a return to man's state in Eden, so that God can dwell with the Israelites as he had with Adam and Eve through the Ark and Tabernacle, which together form a model of the universe ; in later Abrahamic religions this came to be interpreted as Israel being the guardian of God's plan for mankind, to bring " God's creation blessing to mankind " begun in Adam.

The Cappadocian Fathers asserted that God's nature is unknowable to man ; helped to develop the framework of hypostases, or three persons united in a single Godhead ; illustrated how Jesus is the eikon of the Father ; and explained the concept of theosis, the belief that all Christians can be assimilated with God in " imitation of the incarnate Son as the divine model.

Isaac's willingness to follow God's command at the cost of his death has been a model for many Jews who preferred martyrdom to violation of the Jewish law.

Psalm 74 evokes the agon model: it opens with a lament over God's desertion of his people and their tribulations, then asks him to remember his past deeds: " You it was who smashed Sea with your might, who battered the heads of the monsters in the waters ; You it was who crushed the heads of Leviathan, who left them for food for the denizens of the desert ..." In this world-view the seas are primordial forces of disorder, and the work of creation is preceded by a divine combat ( or " theomachy ").

The Christian model within biblical theology sees the concept of revelation as progressive ; each new truth supports, expands, and stands upon former revelations of God's truth like brick laying.

Blumenbach theorised that the Circassians were the closest to God's original model of humanity, and thus " the purest and most beautiful whites were the Circassians ".

In the Catholic model, Christians are transformed by God's action, developing a righteousness of their own.

With de Broglie's suggestion of the existence of electron matter waves in 1924, and for a short time before the full 1926 Schrödinger equation treatment of hydrogen like atom, a Bohr electron " wavelength " could be seen to be a function of its momentum, and thus a Bohr orbiting electron was seen to orbit in a circle at a multiple of its half-wavelength ( this historically incorrect Bohr model is still occasionally taught to students ).

It is deduced from the model existence theorem as follows: if there is no formal proof of a formula then adding its negation to the axioms gives a consisten theory, which has thus a model, so that the formula is not a semantic consequence of the initial theory.

Applying the completeness theorem to this result, gives the existence of a model of T where the formula C < sub > T </ sub > is false.

In fact, the model of any theory containing PA obtained by the systematic construction of the arithmetical model existence theorem, is always non-standard with a non-equivalent provability predicate and a non-equivalent way to interpret its own construction, so that this construction is non-recursive ( as recursive definitions would be unambiguous ).

: This model has been used in general equilibrium theory, particularly to show existence and Pareto efficiency of economic equilibria.

The most successful type of structure prediction, known as homology modeling, relies on the existence of a " template " structure with sequence similarity to the protein being modeled ; structural genomics ' goal is to provide sufficient representation in solved structures to model most of those that remain.

As the faster ROW or " Euro " model was not available in the United States and Canada during the first three years of its existence, a " Competition Group " option was created to allow North American customers to have an S model lookalike with spoilers, 16 " flat disc wheels, sport seats, sport springs and Bilstein shocks.

Other socialists have focused their analyses on the lack of self-management, the continued existence of financial calculation and the existence of a bureaucratic elite based on hierarchical and centralised powers of authority in the Soviet model, leading them to conclude that the Soviet system was not socialist, instead categorising it as bureaucratic collectivism, state capitalism or deformed workers ' states.

Furthermore, cosmological experiments also suggest the existence of dark matter, supposedly composed of fundamental particles outside the scheme of the standard model.

Jung agreed with Freud's model of the unconscious, what Jung called the " personal unconscious ", but he also proposed the existence of a second, far deeper form of the unconscious underlying the personal one.

One problem is that there is currently no theoretical model that explains the earliest moments of the universe's existence ( during the Planck time ) because of a lack of a testable theory of quantum gravity.

This theory might lead to concerns about the existence of free will ( in this model, free will may be an illusion, or at least not unlimited ).

Hoyle later gave Lemaître's model the derisive term of Big Bang, not realizing that Lemaître's model was needed to explain the existence of deuterium and nuclides between helium and carbon, as well as the fundamentally high amount of helium present, not only in stars but also in interstellar gas.

Calculations using a quantum-tunneling model predict the existence of several neutron-rich isotopes of ununoctium with alpha-decay half-lives close to 1 ms.

Their existence was eventually forgotten until the 1870s, when references to the medals in Jefferson's papers were connected to the discovery of Dupré's lead working model.

at MIT found that some business models, as defined by them, indeed performed better than others in a dataset consisting of the largest U. S. firms, in the period 1998 through 2002, while they did not prove whether the existence of a business model mattered.

Calculations using a quantum-tunneling model predict the existence of several isotopes of ununseptium with alpha-decay half-lives up to over 10 ms, culminating at < sup > 296 </ sup > Uus ( the research ends on < sup > 303 </ sup > Uus ).

The model for this new existence was the journey of the Sun.

The free electron model gave improved predictions for the heat capacity of metals, however, it was unable to explain the existence of insulators.

The properties of preon-degenerate matter depend very strongly on the model chosen to describe preons, and the existence of preons is not assumed by the majority of the scientific community, due to conflicts between the preon models originally proposed and experimental data from particle accelerators.