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Exponentially Modified Gaussian Distribution
In probability theory, an exponentially modified Gaussian distribution (EMG, also known as exGaussian distribution) describes the sum of independent normal and exponential random variables. An exGaussian random variable ''Z'' may be expressed as , where ''X'' and ''Y'' are independent, ''X'' is Gaussian with mean ''μ'' and variance ''σ''2, and ''Y'' is exponential of rate ''λ''. It has a characteristic positive skew from the exponential component. It may also be regarded as a weighted function of a shifted exponential with the weight being a function of the normal distribution. Definition The probability density function (pdf) of the exponentially modified normal distribution is :f(x;\mu,\sigma,\lambda) = \frac e^ \operatorname \left(\frac\right), where erfc is the complementary error function defined as :\begin \operatorname(x) & = 1-\operatorname(x) \\ & = \frac \int_x^\infty e^\,dt. \end This density ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
EMG Distribution PDF
EMG may refer to: Medicine and science * Electromyography, a technique for evaluating and recording electrical activity produced by skeletal muscles * Exponentially modified Gaussian distribution, in probability theory * Ɱ, or emg, a symbol used to transcribe a specific sound in the International Phonetic Alphabet Organisations * East Mediterranean Gas Company, an Egyptian pipeline company * EMG, Inc., an American guitar pickup manufacturer * E.M.G. Hand-Made Gramophones, a British gramophone manufacturer * Escape Media Group, Inc., owner of Grooveshark * Essential Media Group, former name of EQ Media Group * Euclid Media Group, an American media company * Executive Music Group, an American record label Other * Eastern Mewahang language * European Masters Games The European Masters Games (EMG) is a multi-sport event, consisting of summer sports, that is held every four years. The European Masters Games are owned by the International Masters Games Association ( IMG ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exponential Decay
A quantity is subject to exponential decay if it decreases at a rate proportional to its current value. Symbolically, this process can be expressed by the following differential equation, where is the quantity and (lambda) is a positive rate called the exponential decay constant, disintegration constant, rate constant, or transformation constant: :\frac = -\lambda N. The solution to this equation (see derivation below) is: :N(t) = N_0 e^, where is the quantity at time , is the initial quantity, that is, the quantity at time . Measuring rates of decay Mean lifetime If the decaying quantity, ''N''(''t''), is the number of discrete elements in a certain set, it is possible to compute the average length of time that an element remains in the set. This is called the mean lifetime (or simply the lifetime), where the exponential time constant, \tau, relates to the decay rate constant, λ, in the following way: :\tau = \frac. The mean lifetime can be looked at as a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exponential Distribution
In probability theory and statistics, the 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. The exponential distribution is not the same as the class of exponential families of distributions. This is a large class of probability distributions that includes the exponential distribution as one of its members, but also includes many other distributions, like the normal, binomial, gamma, and Poisson distributions. Definitions Probability density function The probability density function (pdf) of an exponential distribution is : f(x;\lambda) = \begin \lambda ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compound Probability Distribution
In probability and statistics, a compound probability distribution (also known as a mixture distribution or contagious distribution) is the probability distribution that results from assuming that a random variable is distributed according to some parametrized distribution, with (some of) the parameters of that distribution themselves being random variables. If the parameter is a scale parameter, the resulting mixture is also called a scale mixture. The compound distribution ("unconditional distribution") is the result of marginalizing (integrating) over the ''latent'' random variable(s) representing the parameter(s) of the parametrized distribution ("conditional distribution"). Definition A compound probability distribution is the probability distribution that results from assuming that a random variable X is distributed according to some parametrized distribution F with an unknown parameter \theta that is again distributed according to some other distribution G. The resulting di ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Skew Normal Distribution
In probability theory and statistics, the skew normal distribution is a continuous probability distribution that generalises the normal distribution In 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) = \frac e^ The parameter \mu ... to allow for non-zero skewness. Definition Let \phi(x) denote the Normal distribution, standard normal probability density function :\phi(x)=\frace^ with the cumulative distribution function given by :\Phi(x) = \int_^ \phi(t)\ dt = \frac \left[ 1 + \operatorname \left(\frac\right)\right], where "erf" is the error function. Then the probability density function (pdf) of the skew-normal distribution with parameter \alpha is given by :f(x) = 2\phi(x)\Phi(\alpha x). \, This distribution was first introduced by O'Hagan and Leonard (1976). Alternative forms to this distribution, with the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normal-exponential-gamma Distribution
In probability theory and statistics, the normal-exponential-gamma distribution (sometimes called the NEG distribution) is a three-parameter family of continuous probability distributions. It has a location parameter \mu, scale parameter \theta and a shape parameter k . Probability density function The probability density function (pdf) of the normal-exponential-gamma distribution is proportional to :f(x;\mu, k,\theta) \propto \expD_\left(\frac\right), where ''D'' is a parabolic cylinder function.http://www.newton.ac.uk/programmes/SCB/seminars/121416154.html As for the Laplace distribution, the pdf of the NEG distribution can be expressed as a mixture of normal distributions, :f(x;\mu, k,\theta)=\int_0^\infty\int_0^\infty\ \mathrm(x, \mu, \sigma^2)\mathrm(\sigma^2, \psi)\mathrm(\psi, k, 1/\theta^2) \, d\sigma^2 \, d\psi, where, in this notation, the distribution-names should be interpreted as meaning the density functions of those distributions. Within this scale mixtur ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stochastic Frontier Analysis
Stochastic frontier analysis (SFA) is a method of economic modeling. It has its starting point in the stochastic production frontier models simultaneously introduced by Aigner, Lovell and Schmidt (1977) and Meeusen and Van den Broeck (1977). The ''production frontier model'' without random component can be written as: y_i = f(x_i ;\beta ) \cdot TE_i the best where ''yi'' is the observed scalar output of the producer ''i'', ''i=1,..I, xi'' is a vector of ''N'' inputs used by the producer ''i'', ''f(xi, β)'' is the production frontier, and \beta is a vector of technology parameters to be estimated. ''TEi'' denotes the technical efficiency defined as the ratio of observed output to maximum feasible output. ''TEi = 1'' shows that the ''i-th'' firm obtains the maximum feasible output, while ''TEi < 1'' provides a measure of the shortfall of the observed output from maximum feasible output. A stochastic component that describes random shocks affecting the production process is add ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cell Cycle
The cell cycle, or cell-division cycle, is the series of events that take place in a cell that cause it to divide into two daughter cells. These events include the duplication of its DNA (DNA replication) and some of its organelles, and subsequently the partitioning of its cytoplasm, chromosomes and other components into two daughter cells in a process called cell division. In cells with nuclei ( eukaryotes, i.e., animal, plant, fungal, and protist cells), the cell cycle is divided into two main stages: interphase and the mitotic (M) phase (including mitosis and cytokinesis). During interphase, the cell grows, accumulating nutrients needed for mitosis, and replicates its DNA and some of its organelles. During the mitotic phase, the replicated chromosomes, organelles, and cytoplasm separate into two new daughter cells. To ensure the proper replication of cellular components and division, there are control mechanisms known as cell cycle checkpoints after each of the key steps ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chromatography
In chemical analysis, chromatography is a laboratory technique for the separation of a mixture into its components. The mixture is dissolved in a fluid solvent (gas or liquid) called the ''mobile phase'', which carries it through a system (a column, a capillary tube, a plate, or a sheet) on which a material called the ''stationary phase'' is fixed. Because the different constituents of the mixture tend to have different affinities for the stationary phase and are retained for different lengths of time depending on their interactions with its surface sites, the constituents travel at different apparent velocities in the mobile fluid, causing them to separate. The separation is based on the differential partitioning between the mobile and the stationary phases. Subtle differences in a compound's partition coefficient result in differential retention on the stationary phase and thus affect the separation. Chromatography may be preparative or analytical. The purpose of preparativ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nonparametric Skew
In statistics and probability theory, the nonparametric skew is a statistic occasionally used with random variables that take real values.Arnold BC, Groeneveld RA (1995) Measuring skewness with respect to the mode. The American Statistician 49 (1) 34–38 DOI:10.1080/00031305.1995.10476109Rubio F.J.; Steel M.F.J. (2012) "On the Marshall–Olkin transformation as a skewing mechanism". ''Computational Statistics & Data Analysis''Preprint/ref> It is a measure of the skewness of a random variable's distribution—that is, the distribution's tendency to "lean" to one side or the other of the mean. Its calculation does not require any knowledge of the form of the underlying distribution—hence the name nonparametric. It has some desirable properties: it is zero for any symmetric distribution; it is unaffected by a scale shift; and it reveals either left- or right-skewness equally well. In some statistical samples it has been shown to be less powerfulTabor J (2010) Investiga ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vincent Average
In applied statistics, Vincentization was described by Ratcliff (1979), and is named after biologist S. B. Vincent (1912), who used something very similar to it for constructing learning curves at the beginning of the 1900s. It basically consists of averaging n\geq 2 subjects' estimated or elicited quantile function In probability and statistics, the quantile function, associated with a probability distribution of a random variable, specifies the value of the random variable such that the probability of the variable being less than or equal to that value equ ...s in order to define group quantiles from which F can be constructed. To cast it in its greatest generality, let F_1,\dots, F_n represent arbitrary (empirical or theoretical) distribution functions and define their corresponding quantile functions by : F_i^(\alpha) = \inf\,\quad 0<\alpha\leq 1. The Vincent average of the 's is then computed as : |
Skewness
In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable about its mean. The skewness value can be positive, zero, negative, or undefined. For a unimodal distribution, negative skew commonly indicates that the ''tail'' is on the left side of the distribution, and positive skew indicates that the tail is on the right. In cases where one tail is long but the other tail is fat, skewness does not obey a simple rule. For example, a zero value means that the tails on both sides of the mean balance out overall; this is the case for a symmetric distribution, but can also be true for an asymmetric distribution where one tail is long and thin, and the other is short but fat. Introduction Consider the two distributions in the figure just below. Within each graph, the values on the right side of the distribution taper differently from the values on the left side. These tapering sides are called ''tail ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
