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Bates Distribution
In probability and business statistics, the Bates distribution, named after Grace Bates, is a probability distribution of the mean of a number of statistically independent uniformly distributed random variables on the unit interval. This distribution is related to the uniform, the triangular, and the normal Gaussian distribution, and has applications in broadcast engineering for signal enhancement. The Bates distribution is sometimes confused with the Irwin–Hall distribution, which is the distribution of the sum (not the mean) of ''n'' independent random variables uniformly distributed from 0 to 1. Thus, the two distributions are simply ''versions'' of each other as they only differ in scale. Definition The Bates distribution is the continuous probability distribution of the mean, ''X'', of ''n'' independent, uniformly distributed, random variables on the unit interval, ''Uk'': : X = \frac\sum_^n U_k. The equation defining the probability density function of a Bates ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Independence (probability Theory)
Independence is a fundamental notion in probability theory, as in statistics and the theory of stochastic processes. Two events are independent, statistically independent, or stochastically independent if, informally speaking, the occurrence of one does not affect the probability of occurrence of the other or, equivalently, does not affect the odds. Similarly, two random variables are independent if the realization of one does not affect the probability distribution of the other. When dealing with collections of more than two events, two notions of independence need to be distinguished. The events are called pairwise independent if any two events in the collection are independent of each other, while mutual independence (or collective independence) of events means, informally speaking, that each event is independent of any combination of other events in the collection. A similar notion exists for collections of random variables. Mutual independence implies pairwise independence ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uniform Distribution (continuous)
In probability theory and statistics, the continuous uniform distribution or rectangular distribution is a family of symmetric probability distributions. The distribution describes an experiment where there is an arbitrary outcome that lies between certain bounds. The bounds are defined by the parameters, ''a'' and ''b'', which are the minimum and maximum values. The interval can either be closed (e.g. , b or open (e.g. (a, b)). Therefore, the distribution is often abbreviated ''U'' (''a'', ''b''), where U stands for uniform distribution. The difference between the bounds defines the interval length; all intervals of the same length on the distribution's support are equally probable. It is the maximum entropy probability distribution for a random variable ''X'' under no constraint other than that it is contained in the distribution's support. Definitions Probability density function The probability density function of the continuous uniform distribution is: : f(x)=\begin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Central Limit Theorem
In probability theory, the central limit theorem (CLT) establishes that, in many situations, when independent random variables are summed up, their properly normalized sum tends toward a normal distribution even if the original variables themselves are not normally distributed. The theorem is a key concept in probability theory because it implies that probabilistic and statistical methods that work for normal distributions can be applicable to many problems involving other types of distributions. This theorem has seen many changes during the formal development of probability theory. Previous versions of the theorem date back to 1811, but in its modern general form, this fundamental result in probability theory was precisely stated as late as 1920, thereby serving as a bridge between classical and modern probability theory. If X_1, X_2, \dots, X_n, \dots are random samples drawn from a population with overall mean \mu and finite variance and if \bar_n is the sample mean of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Side Lobe
In antenna engineering, sidelobes are the lobes (local maxima) of the far field radiation pattern of an antenna or other radiation source, that are not the ''main lobe''. The radiation pattern of most antennas shows a pattern of "''lobes''" at various angles, directions where the radiated signal strength reaches a maximum, separated by "''nulls''", angles at which the radiated signal strength falls to zero. This can be viewed as the diffraction pattern of the antenna. In a directional antenna in which the objective is to emit the radio waves in one direction, the lobe in that direction is designed to have a larger field strength than the others; this is the "''main lobe''". The other lobes are called "''sidelobes''", and usually represent unwanted radiation in undesired directions. The sidelobe directly behind the main lobe is called the back lobe. The longer the antenna relative to the radio wavelength, the more lobes its radiation pattern has. In transmitting antenna ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Main Lobe
In a radio antenna's radiation pattern, the main lobe, or main beam, is the lobe containing the higher power. This is the lobe that exhibits the greater field strength. The radiation pattern of most antennas shows a pattern of "''lobes''" at various angles, directions where the radiated signal strength reaches a maximum, separated by "''nulls''", angles at which the radiation falls to zero. In a directional antenna in which the objective is to emit the radio waves in one direction, the lobe in that direction is designed to have higher field strength than the others, so on a graph of the radiation pattern it appears biggest; this is the main lobe. The other lobes are called "''sidelobes''", and usually represent unwanted radiation in undesired directions. The sidelobe in the opposite direction from the main lobe is called the "''backlobe''". The radiation pattern referred to above is usually the horizontal radiation pattern, which is plotted as a function of azimuth about th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Beamwidth
The beam diameter or beam width of an electromagnetic beam is the diameter along any specified line that is perpendicular to the beam axis and intersects it. Since beams typically do not have sharp edges, the diameter can be defined in many different ways. Five definitions of the beam width are in common use: D4σ, 10/90 or 20/80 knife-edge, 1/e2, FWHM, and D86. The beam width can be measured in units of length at a particular plane perpendicular to the beam axis, but it can also refer to the angular width, which is the angle subtended by the beam at the source. The angular width is also called the beam divergence. Beam diameter is usually used to characterize electromagnetic beams in the optical regime, and occasionally in the microwave regime, that is, cases in which the aperture from which the beam emerges is very large with respect to the wavelength. Beam diameter usually refers to a beam of circular cross section, but not necessarily so. A beam may, for example, have an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Short-tailed Distribution
In probability theory and statistics, kurtosis (from el, κυρτός, ''kyrtos'' or ''kurtos'', meaning "curved, arching") is a measure of the "tailedness" of the probability distribution of a real-valued random variable. Like skewness, kurtosis describes a particular aspect of a probability distribution. There are different ways to quantify kurtosis for a theoretical distribution, and there are corresponding ways of estimating it using a sample from a population. Different measures of kurtosis may have different interpretations. The standard measure of a distribution's kurtosis, originating with Karl Pearson, is a scaled version of the fourth moment of the distribution. This number is related to the tails of the distribution, not its peak; hence, the sometimes-seen characterization of kurtosis as "peakedness" is incorrect. For this measure, higher kurtosis corresponds to greater extremity of deviations (or outliers), and not the configuration of data near the mean. It is c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Long Tail
In statistics and business, a long tail of some probability distribution, distributions of numbers is the portion of the distribution having many occurrences far from the "head" or central part of the distribution. The distribution could involve popularities, random numbers of occurrences of events with various probabilities, etc. The term is often used loosely, with no definition or an arbitrary definition, but precise definitions are possible. In statistics, the term ''long-tailed distribution'' has a narrow technical meaning, and is a subtype of heavy-tailed distribution. Intuitively, a distribution is (right) long-tailed if, for any fixed amount, when a quantity exceeds a high level, it almost certainly exceeds it by at least that amount: large quantities are probably even larger. Note that there is no sense of ''the'' "long tail" of a distribution, but only the ''property'' of a distribution being long-tailed. In business, the term ''long tail'' is applied to rank-size dis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Student-t Distribution
In probability and statistics, Student's ''t''-distribution (or simply the ''t''-distribution) is any member of a family of continuous probability distributions that arise when estimating the mean of a normally distributed population in situations where the sample size is small and the population's standard deviation is unknown. It was developed by English statistician William Sealy Gosset under the pseudonym "Student". The ''t''-distribution plays a role in a number of widely used statistical analyses, including Student's ''t''-test for assessing the statistical significance of the difference between two sample means, the construction of confidence intervals for the difference between two population means, and in linear regression analysis. Student's ''t''-distribution also arises in the Bayesian analysis of data from a normal family. If we take a sample of n observations from a normal distribution, then the ''t''-distribution with \nu=n-1 degrees of freedom can be de ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trapezoidal Distribution
In probability theory and statistics, the trapezoidal distribution is a continuous probability distribution whose probability density function graph resembles a trapezoid. Likewise, trapezoidal distributions also roughly resemble mesas or plateaus. Each trapezoidal distribution has a lower bound and an upper bound , where , beyond which no values or events on the distribution can occur (i.e. beyond which the probability is always zero). In addition, there are two sharp bending points (non-differentiable discontinuities) within the probability distribution, which we will call and , which occur between and , such that . The image to the right shows a perfectly linear trapezoidal distribution. However, not all trapezoidal distributions are so precisely shaped. In the standard case, where the middle part of the trapezoid is completely flat, and the side ramps are perfectly linear, all of the values between and will occur with equal frequency, and therefore all such points will ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shape Parameter
In probability theory and statistics, a shape parameter (also known as form parameter) is a kind of numerical parameter of a parametric family of probability distributionsEveritt B.S. (2002) Cambridge Dictionary of Statistics. 2nd Edition. CUP. that is neither a location parameter nor a scale parameter (nor a function of these, such as a rate parameter). Such a parameter must affect the ''shape'' of a distribution rather than simply shifting it (as a location parameter does) or stretching/shrinking it (as a scale parameter does). For example, "peakedness" refers to how round the main peak is. Estimation Many estimators measure location or scale; however, estimators for shape parameters also exist. Most simply, they can be estimated in terms of the higher moments, using the method of moments, as in the ''skewness'' (3rd moment) or ''kurtosis'' (4th moment), if the higher moments are defined and finite. Estimators of shape often involve higher-order statistics (non-linear functi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
