Gaussian functionIn mathematics, a Gaussian function, often simply referred to as a Gaussian, is a function of the base form f(x)=exp(−x2){\displaystyle f(x)=\exp(-x^{2})} and with parametric extension f(x)=aexp(−(x−b)22c2){\displaystyle f(x)=a\exp \left(-{\frac {(x-b)^{2}}{2c^{2}}} ight)} for arbitrary real constants a, b and non-zero c. It is named after the mathematician Carl Friedrich Gauss. The graph of a Gaussian is a characteristic symmetric "bell curve" shape.Cumulative distribution functionProbability that random variable X is less than or equal to xCumulative distribution function for the exponential distributionCumulative distribution function for the normal distributionIn probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variableX{\displaystyle X}, or just distribution function of X{\displaystyle X}, evaluated at x{\displaystyle x}, is the probability that X{\displaystyle X} will take a value less than or equal to x{\displaystyle x}.
Error functionIn mathematics, the error function (also called the Gauss error function), often denoted by 𝐞𝐫𝐟, is the function erf(z)=2π∫0ze−t2dt. The integral here is a complex contour integral which is path-independent because exp(−t2) is holomorphic on the whole complex plane ℂ. In many applications, the function argument is a real number, in which case the function value is also real. In some older texts, the error function is defined without the factor of 2/π.
Risch algorithmMethod for evaluating indefinite integrals In symbolic computation, the Risch algorithm is a method of indefinite integration used in some computer algebra systems to find antiderivatives. It is named after the American mathematician Robert Henry Risch, a specialist in computer algebra who developed it in 1968. The algorithm transforms the problem of integration into a problem in algebra.
Normalizing constantIn probability theory, a normalizing constant or normalizing factor is used to reduce any nonnegative function whose integral is finite to a probability density function. For example, a Gaussian function can be normalized into a probability density function, which gives the standard normal distribution. In Bayes' theorem, a normalizing constant is used to ensure that the sum of all possible hypotheses equals 1.
Partition function (statistical mechanics)In physics, a partition function describes the statistical properties of a system in thermodynamic equilibrium. Partition functions are functions of the thermodynamic state variables, such as the temperature and volume. Most of the aggregate thermodynamic variables of the system, such as the total energy, free energy, entropy, and pressure, can be expressed in terms of the partition function or its derivatives. The partition function is dimensionless.
Normal distributionIn probability theory and 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 f(x)=12πσ2exp(−(x−μ)22σ2). The parameter μ is the mean or expectation of the distribution (and also its median and mode), while the parameter σ2 is the variance.
IntegralIn mathematics, an integral is the continuous analog of a sum, and is used to calculate areas, volumes, and their generalizations. The process of computing an integral, called integration, is one of the two fundamental operations of calculus, along with differentiation. Integration was initially used to solve problems in mathematics and physics, such as finding the area under a curve, or determining displacement from velocity.Elementary functionIn mathematics, an elementary function is a function of a single variable (real or complex) that is typically encountered by beginners. The basic elementary functions are polynomial functions, rational functions, the trigonometric functions, the exponential and logarithm functions, the n-th root, and the inverse trigonometric functions, as well as those functions obtained by addition, multiplication, division, and composition of these.
Abraham de MoivreAbraham de Moivre (French pronunciation: [abʁaam də mwavʁ]; 26 May 1667 – 27 November 1754) was a French mathematician known for de Moivre's formula, a formula that links complex numbers and trigonometry, and for his work on the normal distribution and probability theory. He moved to England at a young age due to the religious persecution of Huguenots in France which reached a climax in 1685 with the Edict of Fontainebleau.
Quantum mechanicsQuantum mechanics, also known as quantum physics, is the fundamental physical theory that describes the behavior of matter and of light; the behaviors it models typically occur at and below the scale of atoms, and have often been described as counterintuitive. Its concepts and methods have been applied across many disciplines, including quantum chemistry, quantum biology, quantum field theory, quantum technology, and quantum information science.
Carl Friedrich GaussJohann Carl Friedrich Gauss ( ; German: Gauß; 30 April 1777 – 23 February 1855) was a German polymath active as a mathematician, astronomer, geodesist, and physicist, who contributed to many fields in mathematics and science. His mathematical contributions spanned the branches of number theory, algebra, analysis, geometry, statistics, and probability. Gauss was director of the Göttingen Observatory in Germany and professor of astronomy from 1807 until his death in 1855.Statistical mechanicsIn physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. Sometimes called statistical physics or statistical thermodynamics, its applications include many problems in a wide variety of fields such as biology, neuroscience, computer science, information theory and sociology.*As an Amazon Associate I earn from qualifying purchases.