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Location Parameter
In statistics, a location family is a class of probability distributions that is parametrized by a scalar- or vector-valued parameter x 0 displaystyle x_ 0 , which determines the "location" or shift of the distribution. Formally, this means that the probability density functions or probability mass functions in this class have the form f x 0 ( x ) = f ( x − x 0 ) . displaystyle f_ x_ 0 (x)=f(x-x_ 0 ). Here, x 0 displaystyle x_ 0 is called the location parameter
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Two-moment Decision Models
In decision theory, economics, and finance, a two-moment decision model is a model that describes or prescribes the process of making decisions in a context in which the decision-maker is faced with random variables whose realizations cannot be known in advance, and in which choices are made based on knowledge of two moments of those random variables
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Probability Density Function
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a 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 equal that sample.[citation needed] In other words, while the absolute likelihood for a continuous random variable to take on any particular value is 0 (since there are 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 equal one sample compared to the other sample. In a more precise sense, the PDF is used to specify the probability of the random variable falling within a particular range of values, as opposed to taking on any one value
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Probability Mass Function
In probability and statistics, a probability mass function (pmf) is a function that gives the probability that a discrete random variable is exactly equal to some value.[1] The probability mass function is often the primary means of defining a discrete probability distribution, and such functions exist for either scalar or multivariate random variables whose domain is discrete. A probability mass function differs from a probability density function (pdf) in that the latter is associated with continuous rather than discrete random variables; the values of the probability density function are not probabilities as such: a pdf must be integrated over an interval to yield a probability.[2] The value of the random variable having the largest probability mass is called the mode.Contents1 Formal definition 2 Measure theoretic formulation 3 Examples3.1 Finite 3.2 Countably infinite4 Multivariate case 5 References 6 Further readingFormal definition[edit] Suppos
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Additive Noise
Additive white Gaussian noise
Gaussian noise
(AWGN) is a basic noise model used in Information theory
Information theory
to mimic the effect of many random processes that occur in nature. The modifiers denote specific characteristics:Additive because it is added to any noise that might be intrinsic to the information system. White refers to the idea that it has uniform power across the frequency band for the information system. It is an analogy to the color white which has uniform emissions at all frequencies in the visible spectrum. Gaussian because it has a normal distribution in the time domain with an average time domain value of zero. Wideband noise comes from many natural sources, such as the thermal vibrations of atoms in conductors (referred to as thermal noise or Johnson-Nyquist noise), shot noise, black body radiation from the earth and other warm objects, and from celestial sources such as the Sun
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Noise
Noise
Noise
is unwanted sound judged to be unpleasant, loud or disruptive to hearing. From a physics standpoint, noise is indistinguishable from sound, as both are vibrations through a medium, such as air or water. The difference arises when the brain receives and perceives a sound.[1][2] In experimental sciences, noise can refer to any random fluctuations of data that hinders perception of an expected signal.[3][4] Acoustic noise is any sound in the acoustic domain, either deliberate (e.g., music or speech) or unintended. In contrast, noise in electronics may not be audible to the human ear and may require instruments for detection.[5] In audio engineering, noise can refer to the unwanted residual electronic noise signal that gives rise to acoustic noise heard as a hiss
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Count Data
In statistics, count data is a statistical data type, a type of data in which the observations can take only the non-negative integer values 0, 1, 2, 3, ... , and where these integers arise from counting rather than ranking
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Statistics
Statistics
Statistics
is a branch of mathematics dealing with the collection, analysis, interpretation, presentation, and organization of data.[1][2] In applying statistics to, for example, a scientific, industrial, or social problem, it is conventional to begin with a statistical population or a statistical model process to be studied. Populations can be diverse topics such as "all people living in a country" or "every atom composing a crystal". Statistics
Statistics
deals with all aspects of data including the planning of data collection in terms of the design of surveys and experiments.[1] See glossary of probability and statistics. When census data cannot be collected, statisticians collect data by developing specific experiment designs and survey samples. Representative sampling assures that inferences and conclusions can reasonably extend from the sample to the population as a whole
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Central Limit Theorem
In probability theory, the central limit theorem (CLT) establishes that, in most situations, when independent random variables are added, their properly normalized sum tends toward a normal distribution (informally a "bell curve") 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. For example, suppose that a sample is obtained containing a large number of observations, each observation being randomly generated in a way that does not depend on the values of the other observations, and that the arithmetic average of the observed values is computed. If this procedure is performed many times, the central limit theorem says that the computed values of the average will be distributed according to a normal distribution
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Moment (mathematics)
In mathematics, a moment is a specific quantitative measure, used in both mechanics and statistics, of the shape of a set of points. If the points represent mass, then the zeroth moment is the total mass, the first moment divided by the total mass is the center of mass, and the second moment is the rotational inertia. If the points represent probability density, then the zeroth moment is the total probability (i.e. one), the first moment is the mean, the second central moment is the variance, the third central moment is the skewness, and the fourth central moment (with normalization and shift) 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 0 to ∞) uniquely determines the distribution (Hausdorff moment problem)
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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 or negative, or undefined. The qualitative interpretation of the skew is complicated and unintuitive. Skew does not refer to the direction the curve appears to be leaning; in fact, the opposite is true. For a unimodal distribution, negative skew indicates that the tail on the left side of the probability density function is longer or fatter than the right side – it does not distinguish these two kinds of shape. Conversely, positive skew indicates that the tail on the right side is longer or fatter than the left side. In cases where one tail is long but the other tail is fat, skewness does not obey a simple rule
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Kurtosis
In probability theory and statistics, kurtosis (from Greek: κυρτός, kyrtos or kurtos, meaning "curved, arching") is a measure of the "tailedness" of the probability distribution of a real-valued random variable. In a similar way to the concept of skewness, kurtosis is a descriptor of the shape of a probability distribution and, just as for skewness, there are different ways of quantifying it for a theoretical distribution and corresponding ways of estimating it from a sample from a population. Depending on the particular measure of kurtosis that is used, there are various interpretations of kurtosis, and of how particular measures should be interpreted. The standard measure of kurtosis, originating with Karl Pearson, is based on a scaled version of the fourth moment of the data or population. This number is related to the tails of the distribution, not its peak;[1] hence, the sometimes-seen characterization as "peakedness" is mistaken
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L-moment
In statistics, L-moments are a sequence of statistics used to summarize the shape of a probability distribution.[1][2][3][4] They are linear combinations of order statistics (L-statistics) analogous to conventional moments, and can be used to calculate quantities analogous to standard deviation, skewness and kurtosis, termed the L-scale, L-skewness and L-kurtosis respectively (the L-mean is identical to the conventional mean). Standardised L-moments are called L-moment ratios and are analogous to standardized moments. Just as for conventional moments, a theoretical distribution has a set of population L-moments
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Index Of Dispersion
In probability theory and statistics, the index of dispersion,[1] dispersion index, coefficient of dispersion, relative variance, or variance-to-mean ratio (VMR), like the coefficient of variation, is a normalized measure of the dispersion of a probability distribution: it is a measure used to quantify whether a set of observed occurrences are clustered or dispersed compared to a standard statistical model. It is defined as the ratio of the variance σ 2 displaystyle sigma ^ 2 to the mean μ displaystyle mu , D = σ 2 μ . displaystyle D= sigma ^ 2 over mu . It is also known as the Fano factor, though this term is sometimes reserved for windowed data (the mean and variance are computed over a subpopulation), where the index of dispersion is used in the special case where the window is infinite
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Grouped Data
Grouped data are data formed by aggregating individual observations of a variable into groups, so that a frequency distribution of these groups serves as a convenient means of summarizing or analyzing the data.Contents1 Example 2 Mean
Mean
of grouped data 3 See also 4 Notes 5 ReferencesExample[edit] The idea of grouped data can be illustrated by considering the following raw dataset: Table 1: Time taken (in seconds) by a group of students to answer a simple math question20 25 24 33 1326 8 19 31 1116 21 17 11 3414 15 21 18 17The above data can be grouped in order to construct a frequency distribution in any of several ways. One method is to use intervals as a basis. The smallest value in the above data is 8 and the largest is 34. The interval from 8 to 34 is broken up into smaller subintervals (called class intervals). For each class interval, the amount of data items falling in this interval is counted
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Frequency Distribution
In statistics, a frequency distribution is a list, table or graph that displays the frequency of various outcomes in a sample.[1] Each entry in the table contains the frequency or count of the occurrences of values within a particular group or interval, and in this way, the table summarizes the distribution of values in the sample.Contents1 Univariate frequency tables 2 Construction of frequency distributions 3 Joint frequency distributions 4 Applications 5 See also 6 Notes 7 External linksUnivariate frequency tables[edit] Here is an example of a univariate (i.e. single variable) frequency table. The frequency of each response to a survey question is depicted.Rank Degree of agreement Number1 Strongly agree 202 Agree somewhat 303 Not sure 204 Disagree somewhat 155 Strongly disagree 15A different tabulation scheme aggregates values into bins such that each bin encompasses a range of values
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