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Modified Lognormal Power-law Distribution
The modified lognormal power-law (MLP) function is a three parameter function that can be used to model data that have characteristics of a log-normal distribution and a power law behavior. It has been used to model the functional form of the initial mass function (IMF). Unlike the other functional forms of the IMF, the MLP is a single function with no joining conditions. Functional form The closed form of the probability density function of the MLP is as follows: :\begin f(m)= \frac \exp\left(\alpha \mu _0+ \frac\right) m^ \text\left( \frac\left(\alpha \sigma _0 -\frac\right)\right),\ m \in [0,\infty) \end where \begin \alpha = \frac \end is the asymptotic power-law index of the distribution. Here \mu_0 and \sigma_0^2 are the mean and variance, respectively, of an underlying lognormal distribution from which the MLP is derived. Mathematical properties Following are the few mathematical properties of the MLP distribution: Cumulative distribution The MLP cumulative distrib ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Log-normal Distribution
In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed. Thus, if the random variable is log-normally distributed, then has a normal distribution. Equivalently, if has a normal distribution, then the exponential function of , , has a log-normal distribution. A random variable which is log-normally distributed takes only positive real values. It is a convenient and useful model for measurements in exact and engineering sciences, as well as medicine, economics and other topics (e.g., energies, concentrations, lengths, prices of financial instruments, and other metrics). The distribution is occasionally referred to as the Galton distribution or Galton's distribution, after Francis Galton. The log-normal distribution has also been associated with other names, such as McAlister, Gibrat and Cobb–Douglas. A log-normal process is the statistical realization of the multipl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Power Law
In statistics, a power law is a Function (mathematics), functional relationship between two quantities, where a Relative change and difference, relative change in one quantity results in a proportional relative change in the other quantity, independent of the initial size of those quantities: one quantity varies as a Exponentiation, power of another. For instance, considering the area of a square in terms of the length of its side, if the length is doubled, the area is multiplied by a factor of four. Empirical examples The distributions of a wide variety of physical, biological, and man-made phenomena approximately follow a power law over a wide range of magnitudes: these include the sizes of craters on the moon and of solar flares, the foraging pattern of various species, the sizes of activity patterns of neuronal populations, the frequencies of words in most languages, frequencies of family names, the species richness in clades of organisms, the sizes of power outages, volcanic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Initial Mass Function
In astronomy, the initial mass function (IMF) is an empirical function that describes the initial Frequency distribution, distribution of masses for a population of stars. The IMF is an output of the process of star formation. The IMF is often given as a Probability distribution, probability distribution function (PDF) for the mass at which a star enters the main sequence (begins nuclear fusion, hydrogen fusion). The distribution function can then be used to construct the mass distribution (the histogram of stellar masses) of a population of stars. It differs from the ''present day mass function'' (PDMF), the current distribution of masses of stars, due to the evolution and death of stars which occurs at different rates for different masses as well as dynamical mixing in some populations. The properties and evolution of a star are closely related to its mass, so the IMF is an important diagnostic tool for astronomers studying large quantities of stars. For example, the initial ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cumulative Distribution Function
In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. Every probability distribution supported on the real numbers, discrete or "mixed" as well as continuous, is uniquely identified by an ''upwards continuous'' ''monotonic increasing'' cumulative distribution function F : \mathbb R \rightarrow ,1/math> satisfying \lim_F(x)=0 and \lim_F(x)=1. In the case of a scalar continuous distribution, it gives the area under the probability density function from minus infinity to x. Cumulative distribution functions are also used to specify the distribution of multivariate random variables. Definition The cumulative distribution function of a real-valued random variable X is the function given by where the right-hand side represents the probability that the random variable X takes on a value less tha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Expectation Value
In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a large number of independently selected outcomes of a random variable. The expected value of a random variable with a finite number of outcomes is a weighted average of all possible outcomes. In the case of a continuum of possible outcomes, the expectation is defined by integration. In the axiomatic foundation for probability provided by measure theory, the expectation is given by Lebesgue integration. The expected value of a random variable is often denoted by , , or , with also often stylized as or \mathbb. History The idea of the expected value originated in the middle of the 17th century from the study of the so-called problem of points, which seeks to divide the stakes ''in a fair way'' between two players, who have to en ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Raw Moment
In mathematics, the moments of a function are certain quantitative measures related to the shape of the function's graph. If the function represents mass density, then the zeroth moment is the total mass, the first moment (normalized by total mass) is the center of mass, and the second moment is the moment of inertia. If the function is a probability distribution, then the first moment is the expected value, the second central moment is the variance, the third standardized moment is the skewness, and the fourth standardized moment is the kurtosis. The mathematical concept is closely related to the concept of moment in physics. For a distribution of mass or probability on a bounded interval, the collection of all the moments (of all orders, from to ) uniquely determines the distribution (Hausdorff moment problem). The same is not true on unbounded intervals (Hamburger moment problem). In the mid-nineteenth century, Pafnuty Chebyshev became the first person to think systema ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
