Factorial momentIn probability theory, the factorial moment is a mathematical quantity defined as the expectation or average of the falling factorial of a random variable. Factorial moments are useful for studying non-negative integer-valued random variables, and arise in the use of probability-generating functions to derive the moments of discrete random variables. Factorial moments serve as analytic tools in the mathematical field of combinatorics, which is the study of discrete mathematical structures.
Integral transformIn mathematics, an integral transform maps a function from its original function space into another function space via integration, where some of the properties of the original function might be more easily characterized and manipulated than in the original function space. The transformed function can generally be mapped back to the original function space using the inverse transform. An integral transform is any transform of the following form: The input of this transform is a function , and the output is another function .
Frequentist probabilityFrequentist probability or frequentism is an interpretation of probability; it defines an event's probability as the limit of its relative frequency in many trials (the long-run probability). Probabilities can be found (in principle) by a repeatable objective process (and are thus ideally devoid of opinion). The continued use of frequentist methods in scientific inference, however, has been called into question. The development of the frequentist account was motivated by the problems and paradoxes of the previously dominant viewpoint, the classical interpretation.
Wigner semicircle distributionThe Wigner semicircle distribution, named after the physicist Eugene Wigner, is the probability distribution on [−R, R] whose probability density function f is a scaled semicircle (i.e., a semi-ellipse) centered at (0, 0): for −R ≤ x ≤ R, and f(x) = 0 if |x| > R. The parameter R is commonly referred to as the "radius" parameter of the distribution. The Wigner distribution also coincides with a scaled beta distribution. That is, if Y is a beta-distributed random variable with parameters α = β = 3/2, then the random variable X = 2RY – R exhibits a Wigner semicircle distribution with radius R.
Binomial typeIn mathematics, a polynomial sequence, i.e., a sequence of polynomials indexed by non-negative integers in which the index of each polynomial equals its degree, is said to be of binomial type if it satisfies the sequence of identities Many such sequences exist. The set of all such sequences forms a Lie group under the operation of umbral composition, explained below. Every sequence of binomial type may be expressed in terms of the Bell polynomials. Every sequence of binomial type is a Sheffer sequence (but most Sheffer sequences are not of binomial type).
Probability interpretationsThe word probability has been used in a variety of ways since it was first applied to the mathematical study of games of chance. Does probability measure the real, physical, tendency of something to occur, or is it a measure of how strongly one believes it will occur, or does it draw on both these elements? In answering such questions, mathematicians interpret the probability values of probability theory. There are two broad categories of probability interpretations which can be called "physical" and "evidential" probabilities.
Itô's lemmaIn mathematics, Itô's lemma or Itô's formula (also called the Itô-Doeblin formula, especially in the French literature) is an identity used in Itô calculus to find the differential of a time-dependent function of a stochastic process. It serves as the stochastic calculus counterpart of the chain rule. It can be heuristically derived by forming the Taylor series expansion of the function up to its second derivatives and retaining terms up to first order in the time increment and second order in the Wiener process increment.
CT scanA computed tomography scan (usually abbreviated to CT scan; formerly called computed axial tomography scan or CAT scan) is a medical imaging technique used to obtain detailed internal images of the body. The personnel that perform CT scans are called radiographers or radiology technologists. CT scanners use a rotating X-ray tube and a row of detectors placed in a gantry to measure X-ray attenuations by different tissues inside the body.
Itô calculusItô calculus, named after Kiyosi Itô, extends the methods of calculus to stochastic processes such as Brownian motion (see Wiener process). It has important applications in mathematical finance and stochastic differential equations. The central concept is the Itô stochastic integral, a stochastic generalization of the Riemann–Stieltjes integral in analysis. The integrands and the integrators are now stochastic processes: where H is a locally square-integrable process adapted to the filtration generated by X , which is a Brownian motion or, more generally, a semimartingale.
PET-CTPositron emission tomography–computed tomography (better known as PET-CT or PET/CT) is a nuclear medicine technique which combines, in a single gantry, a positron emission tomography (PET) scanner and an x-ray computed tomography (CT) scanner, to acquire sequential images from both devices in the same session, which are combined into a single superposed () image. Thus, functional imaging obtained by PET, which depicts the spatial distribution of metabolic or biochemical activity in the body can be more precisely aligned or correlated with anatomic imaging obtained by CT scanning.
