Exponential familyIn probability and statistics, an exponential family is a parametric set of probability distributions of a certain form, specified below. This special form is chosen for mathematical convenience, including the enabling of the user to calculate expectations, covariances using differentiation based on some useful algebraic properties, as well as for generality, as exponential families are in a sense very natural sets of distributions to consider. The term exponential class is sometimes used in place of "exponential family", or the older term Koopman–Darmois family.
Proof calculusIn mathematical logic, a proof calculus or a proof system is built to prove statements. A proof system includes the components: Language: The set L of formulas admitted by the system, for example, propositional logic or first-order logic. Rules of inference: List of rules that can be employed to prove theorems from axioms and theorems. Axioms: Formulas in L assumed to be valid. All theorems are derived from axioms. Usually a given proof calculus encompasses more than a single particular formal system, since many proof calculi are under-determined and can be used for radically different logics.
Exponential distributionIn probability theory and statistics, the exponential distribution or negative exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate. It is a particular case of the gamma distribution. It is the continuous analogue of the geometric distribution, and it has the key property of being memoryless. In addition to being used for the analysis of Poisson point processes it is found in various other contexts.
Constructive proofIn mathematics, a constructive proof is a method of proof that demonstrates the existence of a mathematical object by creating or providing a method for creating the object. This is in contrast to a non-constructive proof (also known as an existence proof or pure existence theorem), which proves the existence of a particular kind of object without providing an example. For avoiding confusion with the stronger concept that follows, such a constructive proof is sometimes called an effective proof.
Natural exponential familyIn probability and statistics, a natural exponential family (NEF) is a class of probability distributions that is a special case of an exponential family (EF). The natural exponential families (NEF) are a subset of the exponential families. A NEF is an exponential family in which the natural parameter η and the natural statistic T(x) are both the identity. A distribution in an exponential family with parameter θ can be written with probability density function (PDF) where and are known functions.
Romanian alphabetThe Romanian alphabet is a variant of the Latin alphabet used for writing the Romanian language. It is a modification of the classical Latin alphabet and consists of 31 letters, five of which (Ă, Â, Î, Ș, and Ț) have been modified from their Latin originals for the phonetic requirements of the language: The letters Q (chiu), W (dublu v), and Y (igrec or i grec, meaning "Greek i") were formally introduced in the Romanian alphabet in 1982, although they had been used earlier.
Normal distributionIn statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is The parameter is the mean or expectation of the distribution (and also its median and mode), while the parameter is its standard deviation. The variance of the distribution is . A random variable with a Gaussian distribution is said to be normally distributed, and is called a normal deviate.
Confidence intervalIn frequentist statistics, a confidence interval (CI) is a range of estimates for an unknown parameter. A confidence interval is computed at a designated confidence level; the 95% confidence level is most common, but other levels, such as 90% or 99%, are sometimes used. The confidence level, degree of confidence or confidence coefficient represents the long-run proportion of CIs (at the given confidence level) that theoretically contain the true value of the parameter; this is tantamount to the nominal coverage probability.
CircumflexThe circumflex () is a diacritic in the Latin and Greek scripts that is also used in the written forms of many languages and in various romanization and transcription schemes. It received its English name from circumflexus "bent around"a translation of the περισπωμένη (perispōménē). The circumflex in the Latin script is chevron-shaped (), while the Greek circumflex may be displayed either like a tilde () or like an inverted breve (). For the most commonly encountered uses of the accent in the Latin alphabet, precomposed characters are available.
Yo (Cyrillic)Yo, Jo or Io (Ё ё; italics: Ё ё; jɵ) is a letter of the Cyrillic script. In Unicode, the letter is named CYRILLIC CAPITAL/SMALL LETTER IO. In English, the letter Yo is romanized using the Latin ë (according to the ALA–LC and British Standards), ë (yë word-initially) (BGN/PCGN) or yo/jo (orthographic transcription) for Russian, and as i͡o (ALA–LC), yo (BGN/PCGN), or ë (BSI) for Belarusian. In international systems, Yo is romanized as ë (ISO 9). It was derived from the Cyrillic letter Ye (Е е).
Proof assistantIn computer science and mathematical logic, a proof assistant or interactive theorem prover is a software tool to assist with the development of formal proofs by human-machine collaboration. This involves some sort of interactive proof editor, or other interface, with which a human can guide the search for proofs, the details of which are stored in, and some steps provided by, a computer. A recent effort within this field is making these tools use artificial intelligence to automate the formalization of ordinary mathematics.
TheoremIn mathematics, a theorem is a statement that has been proved, or can be proved. The proof of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems. In mainstream mathematics, the axioms and the inference rules are commonly left implicit, and, in this case, they are almost always those of Zermelo–Fraenkel set theory with the axiom of choice (ZFC), or of a less powerful theory, such as Peano arithmetic.
