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In mathematics, the second moment method is a technique used in probability theory and analysis to show that a random variable has positive probability of being positive. More generally, the "moment method" consists of bounding the probability that a random variable fluctuates far from its mean, by using its moments.
All these extensions are also called normal or Gaussian laws, so a certain ambiguity in names exists. The multivariate normal distribution describes the Gaussian law in the k-dimensional Euclidean space. A vector X ∈ R k is multivariate-normally distributed if any linear combination of its components Σ k j=1 a j X j has a (univariate) normal ...
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.
As its name implies, the moment-generating function can be used to compute a distribution’s moments: the nth moment about 0 is the nth derivative of the moment-generating function, evaluated at 0. In addition to real-valued distributions (univariate distributions), moment-generating functions can be defined for vector- or matrix-valued random ...
The equidensity contours of a non-singular multivariate normal distribution are ellipsoids (i.e. affine transformations of hyperspheres) centered at the mean. [29] Hence the multivariate normal distribution is an example of the class of elliptical distributions.
The first cumulant is the expected value; the second and third cumulants are respectively the second and third central moments (the second central moment is the variance); but the higher cumulants are neither moments nor central moments, but rather more complicated polynomial functions of the moments.
The above is obtained using a second order approximation, following the method used in estimating the first moment. It will be a poor approximation in cases where f ( X ) {\displaystyle f(X)} is highly non-linear.
In this case the distribution cannot be interpreted as an untruncated normal conditional on < <, of course, but can still be interpreted as a maximum-entropy distribution with first and second moments as constraints, and has an additional peculiar feature: it presents two local maxima instead of one, located at = and =.