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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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Boxplot
In descriptive statistics, a box plot or boxplot is a method for graphically depicting groups of numerical data through their quartiles. Box plots may also have lines extending vertically from the boxes (whiskers) indicating variability outside the upper and lower quartiles, hence the terms box-and-whisker plot and box-and-whisker diagram. Outliers may be plotted as individual points. Box plots are non-parametric: they display variation in samples of a statistical population without making any assumptions of the underlying statistical distribution. The spacings between the different parts of the box indicate the degree of dispersion (spread) and skewness in the data, and show outliers. In addition to the points themselves, they allow one to visually estimate various L-estimators, notably the interquartile range, midhinge, range, mid-range, and trimean. Box plots can be drawn either horizontally or vertically. Box plots received their name from the box in the middle
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Rademacher Distribution
In probability theory and statistics, the Rademacher distribution (which is named after Hans Rademacher) is a discrete probability distribution where a random variate X has a 50% chance of being +1 and a 50% chance of being -1.[1] A series of Rademacher distributed variables can be regarded as a simple symmetrical random walk where the step size is 1.Contents1 Mathematical formulation 2 Van Zuijlen's bound 3 Bounds on sums 4 Applications 5 Related distributions 6 ReferencesMathematical formulation[edit] The probability mass function of this distribution is f ( k ) = 1 / 2 if  k = − 1 , 1 / 2 if 
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Almost Everywhere
In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of almost everywhere is a companion notion to the concept of measure zero. In the subject of probability, which is largely based in measure theory, the notion is referred to as almost surely. More specifically, a property holds almost everywhere if the set of elements for which the property does not hold is a set of measure zero (Halmos 1974), or equivalently if the set of elements for which the property holds is conull. In cases where the measure is not complete, it is sufficient that the set is contained within a set of measure zero. When discussing sets of real numbers, the Lebesgue measure
Lebesgue measure
is assumed unless otherwise stated. The term almost everywhere is abbreviated a.e.; in older literature p.p
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Uniform Distribution (continuous)
In probability theory and statistics, the continuous uniform distribution or rectangular distribution is a family of symmetric probability distributions such that for each member of the family, all intervals of the same length on the distribution's support are equally probable. The support is defined by the two parameters, a and b, which are its minimum and maximum values. The distribution is often abbreviated U(a,b)
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Expected Value
In probability theory, the expected value of a random variable, intuitively, is the long-run average value of repetitions of the experiment it represents. For example, the expected value in rolling a six-sided dice is 3.5, because the average of all the numbers that come up in an extremely large number of rolls is close to 3.5. Less roughly, the law of large numbers states that the arithmetic mean of the values almost surely converges to the expected value as the number of repetitions approaches infinity. The expected value is also known as the expectation, mathematical expectation, EV, average, mean value, mean, or first moment. More practically, the expected value of a discrete random variable is the probability-weighted average of all possible values. In other words, each possible value the random variable can assume is multiplied by its probability of occurring, and the resulting products are summed to produce the expected value
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Cantor Distribution
The Cantor distribution
Cantor distribution
is the probability distribution whose cumulative distribution function is the Cantor function. This distribution has neither a probability density function nor a probability mass function, since although its cumulative distribution function is a continuous function, the distribution is not absolutely continuous with respect to Lebesgue measure, nor does it have any point-masses
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Absolute Continuity
In calculus, absolute continuity is a smoothness property of functions that is stronger than continuity and uniform continuity. The notion of absolute continuity allows one to obtain generalizations of the relationship between the two central operations of calculus—differentiation and integration—expressed by the fundamental theorem of calculus in the framework of Riemann integration. Such generalizations are often formulated in terms of Lebesgue
Lebesgue
integration. For real-valued functions on the real line two interrelated notions appear: absolute continuity of functions and absolute continuity of measures. These two notions are generalized in different directions
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Derivative
The derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. For example, the derivative of the position of a moving object with respect to time is the object's velocity: this measures how quickly the position of the object changes when time advances. The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point. The tangent line is the best linear approximation of the function near that input value
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Measure Zero
In set theory, a null set N ⊂ R displaystyle Nsubset mathbb R is a set that can be covered by a countable union of intervals of arbitrarily small total length. The notion of null set in set theory anticipates the development of Lebesgue measure
Lebesgue measure
since a null set necessarily has measure zero
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Statistical Physics
Statistical physics
Statistical physics
is a branch of physics that uses methods of probability theory and statistics, and particularly the mathematical tools for dealing with large populations and approximations, in solving physical problems. It can describe a wide variety of fields with an inherently stochastic nature. Its applications include many problems in the fields of physics, biology, chemistry, neurology, and even some social sciences, such as sociology.[citation needed] Its main purpose is to clarify the properties of matter in aggregate, in terms of physical laws governing atomic motion.[1] In particular, statistical mechanics develops the phenomenological results of thermodynamics from a probabilistic examination of the underlying microscopic systems
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Dirac Delta Function
In mathematics, the Dirac delta function, or δ function, is a generalized function, or distribution that was historically introduced by the physicist Paul Dirac
Paul Dirac
for modelling the density of an idealized point mass or point charge, as a function that is equal to zero everywhere except for zero and whose integral over the entire real line is equal to one.[1][2][3] As there is no function that has these properties, the computations that were done by the theoretical physicists appeared to mathematicians as nonsense, until the introduction of distributions by Laurent Schwartz, for formalizing and validating mathematically these computations
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Mean
In mathematics, mean has several different definitions depending on the context. In probability and statistics, population mean and expected value are used synonymously to refer to one measure of the central tendency either of a probability distribution or of the random variable characterized by that distribution.[1] In the case of a discrete probability distribution of a random variable X, the mean is equal to the sum over every possible value weighted by the probability of that value; that is, it is computed by taking the product of each possible value x of X and its probability P(x), and then adding all these products together, giving μ = ∑ x P ( x ) displaystyle mu =sum xP(x) .[2] An analogous formula applies to the case of a continuous probability distribution
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Counting Measure
In mathematics, the counting measure is an intuitive way to put a measure on any set: the "size" of a subset is taken to be: the number of elements in the subset if the subset has finitely many elements, and ∞ if the subset is infinite.[1] The counting measure can be defined on any measurable set, but is mostly used on countable sets.[1] In formal notation, we can make any set X into a measurable space by taking the sigma-algebra Σ displaystyle Sigma of measurable subsets to consist of all subsets of X displaystyle X
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Variance
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its mean. Informally, it measures how far a set of (random) numbers are spread out from their average value. Variance
Variance
has a central role in statistics, where some ideas that use it include descriptive statistics, statistical inference, hypothesis testing, goodness of fit, and Monte Carlo sampling. Variance
Variance
is an important tool in the sciences, where statistical analysis of data is common
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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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