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Quantile Function
In probability and statistics, the quantile function is a function Q: ,1\mapsto \mathbb which maps some probability x \in ,1/math> of a random variable v to the value of the variable y such that P(v\leq y) = x according to its probability distribution. In other words, the function returns the value of the variable below which the specified cumulative probability is contained. For example, if the distribution is a standard normal distribution then Q(0.5) will return 0 as 0.5 of the probability mass is contained below 0. The quantile function is also called the percentile function (after the percentile), percent-point function, inverse cumulative distribution function (after the cumulative distribution function or c.d.f.) or inverse distribution function. Definition Strictly increasing distribution function With reference to a continuous and strictly increasing cumulative distribution function (c.d.f.) F_X\colon \mathbb \to ,1/math> of a random variable , the quantile function ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Probit Plot
In probability theory and statistics, the probit function is the quantile function associated with the standard normal distribution. It has applications in data analysis and machine learning, in particular Q–Q plot, exploratory statistical graphics and specialized probit model, regression modeling of binary response variables. Mathematically, the probit is the inverse function, inverse of the cumulative distribution function of the standard normal distribution, which is denoted as \Phi(z), so the probit is defined as :\operatorname(p) = \Phi^(p) \quad \text \quad p \in (0,1). Largely because of the central limit theorem, the standard normal distribution plays a fundamental role in probability theory and statistics. If we consider the familiar fact that the standard normal distribution places 95% of probability between −1.96 and 1.96 and is symmetric around zero, it follows that :\Phi(-1.96) = 0.025 = 1-\Phi(1.96).\,\! The probit function gives the 'inverse' computation, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Probability Density Function
In probability theory, a probability density function (PDF), density function, or density of an absolutely continuous random variable, is a Function (mathematics), function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can be interpreted as providing a ''relative likelihood'' that the value of the random variable would be equal to that sample. Probability density is the probability per unit length, in other words, while the ''absolute likelihood'' for a continuous random variable to take on any particular value is 0 (since there is an infinite set of possible values to begin with), the value of the PDF at two different samples can be used to infer, in any particular draw of the random variable, how much more likely it is that the random variable would be close to one sample compared to the other sample. More precisely, the PDF is used to specify the probability of the random variable falling ''within ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Closed-form Expression
In mathematics, an expression or equation is in closed form if it is formed with constants, variables, and a set of functions considered as ''basic'' and connected by arithmetic operations (, and integer powers) and function composition. Commonly, the basic functions that are allowed in closed forms are ''n''th root, exponential function, logarithm, and trigonometric functions. However, the set of basic functions depends on the context. For example, if one adds polynomial roots to the basic functions, the functions that have a closed form are called elementary functions. The ''closed-form problem'' arises when new ways are introduced for specifying mathematical objects, such as limits, series, and integrals: given an object specified with such tools, a natural problem is to find, if possible, a ''closed-form expression'' of this object; that is, an expression of this object in terms of previous ways of specifying it. Example: roots of polynomials The quadratic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Numerical Methods
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods that attempt to find approximate solutions of problems rather than the exact ones. Numerical analysis finds application in all fields of engineering and the physical sciences, and in the 21st century also the life and social sciences like economics, medicine, business and even the arts. Current growth in computing power has enabled the use of more complex numerical analysis, providing detailed and realistic mathematical models in science and engineering. Examples of numerical analysis include: ordinary differential equations as found in celestial mechanics (predicting the motions of planets, stars and galaxies), numerical linear algebra in data analysis, and stochastic differential equations and Markov chains for simulating living cells in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Monte Carlo Methods In Finance
Monte Carlo methods are used in corporate finance and mathematical finance to value and analyze (complex) instruments, portfolios and investments by simulating the various sources of uncertainty affecting their value, and then determining the distribution of their value over the range of resultant outcomes. This is usually done by help of stochastic asset models. The advantage of Monte Carlo methods over other techniques increases as the dimensions (sources of uncertainty) of the problem increase. Monte Carlo methods were first introduced to finance in 1964 by David B. Hertz through his ''Harvard Business Review'' article, discussing their application in Corporate Finance. In 1977, Phelim Boyle pioneered the use of simulation in derivative valuation in his seminal ''Journal of Financial Economics'' paper. This article discusses typical financial problems in which Monte Carlo methods are used. It also touches on the use of so-called "quasi-random" methods such as the use of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Copula (statistics)
In probability theory and statistics, a copula is a multivariate cumulative distribution function for which the marginal probability distribution of each variable is Uniform distribution (continuous), uniform on the interval [0, 1]. Copulas are used to describe / model the Dependent and independent variables, dependence (inter-correlation) between random variables. Their name, introduced by applied mathematician Abe Sklar in 1959, comes from the Latin for "link" or "tie", similar but only metaphoricly related to grammatical Copula (linguistics), copulas in linguistics. Copulas have been used widely in quantitative finance to model and minimize tail risk and Portfolio optimization, portfolio-optimization applications. Sklar's theorem states that any multivariate cumulative distribution function#Multivariate case, joint distribution can be written in terms of univariate marginal distribution functions and a copula which describes the dependence structure between the va ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Multivariate Analysis
Multivariate statistics is a subdivision of statistics encompassing the simultaneous observation and analysis of more than one outcome variable, i.e., '' multivariate random variables''. Multivariate statistics concerns understanding the different aims and background of each of the different forms of multivariate analysis, and how they relate to each other. The practical application of multivariate statistics to a particular problem may involve several types of univariate and multivariate analyses in order to understand the relationships between variables and their relevance to the problem being studied. In addition, multivariate statistics is concerned with multivariate probability distributions, in terms of both :*how these can be used to represent the distributions of observed data; :*how they can be used as part of statistical inference, particularly where several different quantities are of interest to the same analysis. Certain types of problems involving multivariate da ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Computational Finance
Computational finance is a branch of applied computer science that deals with problems of practical interest in finance.Rüdiger U. Seydel, ''Tools for Computational Finance'', Springer; 3rd edition (May 11, 2006) 978-3540279235 Some slightly different definitions are the study of data and algorithms currently used in finance and the mathematics of computer programs that realize financial mathematical model, models or systems.Cornelis A. Los, ''Computational Finance'' World Scientific Pub Co Inc (December 2000) Computational finance emphasizes practical numerical analysis, numerical methods rather than mathematical proofs and focuses on techniques that apply directly to economics, economic analyses.Mario J. Miranda and Paul L. Fackler, ''Applied Computational Economics and Finance'', The MIT Press (September 16, 2002) It is an interdisciplinary field between mathematical finance and numerical analysis, numerical methods.Omur Ugur, ''Introduction to Computational Finance'', Imper ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Pseudorandom Number
A pseudorandom sequence of numbers is one that appears to be statistically random, despite having been produced by a completely deterministic and repeatable process. Pseudorandom number generators are often used in computer programming, as traditional sources of randomness available to humans (such as rolling dice) rely on physical processes not readily available to computer programs, although developments in hardware random number generator technology have challenged this. Background The generation of random numbers has many uses, such as for random sampling, Monte Carlo methods, board games, or gambling. In physics, however, most processes, such as gravitational acceleration, are deterministic, meaning that they always produce the same outcome from the same starting point. Some notable exceptions are radioactive decay and quantum measurement, which are both modeled as being truly random processes in the underlying physics. Since these processes are not practical sources of r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Quantile
In statistics and probability, quantiles are cut points dividing the range of a probability distribution into continuous intervals with equal probabilities or dividing the observations in a sample in the same way. There is one fewer quantile than the number of groups created. Common quantiles have special names, such as '' quartiles'' (four groups), '' deciles'' (ten groups), and '' percentiles'' (100 groups). The groups created are termed halves, thirds, quarters, etc., though sometimes the terms for the quantile are used for the groups created, rather than for the cut points. -quantiles are values that partition a finite set of values into subsets of (nearly) equal sizes. There are partitions of the -quantiles, one for each integer satisfying . In some cases the value of a quantile may not be uniquely determined, as can be the case for the median (2-quantile) of a uniform probability distribution on a set of even size. Quantiles can also be applied to continuous di ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Statistical Significance
In statistical hypothesis testing, a result has statistical significance when a result at least as "extreme" would be very infrequent if the null hypothesis were true. More precisely, a study's defined significance level, denoted by \alpha, is the probability of the study rejecting the null hypothesis, given that the null hypothesis is true; and the p-value, ''p''-value of a result, ''p'', is the probability of obtaining a result at least as extreme, given that the null hypothesis is true. The result is said to be ''statistically significant'', by the standards of the study, when p \le \alpha. The significance level for a study is chosen before data collection, and is typically set to 5% or much lower—depending on the field of study. In any experiment or Observational study, observation that involves drawing a Sampling (statistics), sample from a Statistical population, population, there is always the possibility that an observed effect would have occurred due to sampling error al ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Percentage Point
A percentage point or percent point is the unit (measurement), unit for the difference (mathematics), arithmetic difference between two percentages. For example, moving up from 40 percent to 44 percent is an increase of 4 percentage points (although it is a 10-percent increase in the quantity being measured, if the total amount remains the same). In written text, the unit (the percentage point) is usually either written out, or abbreviated as ''pp'', ''p.p.'', or ''%pt.'' to avoid confusion with percentage increase or decrease in the actual quantity. After the first occurrence, some writers abbreviate by using just "point" or "points". Differences between percentages and percentage points Consider the following hypothetical example: In 1980, 50 percent of the population smoked, and in 1990 only 40 percent of the population smoked. One can thus say that from 1980 to 1990, the prevalence of smoking decreased by 10 ''percentage points'' (or by 10 percent of the population) or by ''20 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |