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The first central moment μ 1 is 0 (not to be confused with the first raw moment or the expected value μ). The second central moment μ 2 is called the variance, and is usually denoted σ 2, where σ represents the standard deviation. The third and fourth central moments are used to define the standardized moments which are used to define ...
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.
For any non-negative integer , the plain central moments are: [25] [()] = {()!! Here !! denotes the double factorial, that is, the product of all numbers from to 1 that have the same parity as . The central absolute moments coincide with plain moments for all even orders, but are nonzero for odd orders.
In probability theory and statistics, a standardized moment of a probability distribution is a moment (often a higher degree central moment) that is normalized, typically by a power of the standard deviation, rendering the moment scale invariant. The shape of different probability distributions can be compared using standardized moments. [1]
In image processing, computer vision and related fields, an image moment is a certain particular weighted average of the image pixels' intensities, or a function of such moments, usually chosen to have some attractive property or interpretation. Image moments are useful to describe objects after segmentation.
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 kurtosis κ is defined to be the normalized fourth central moment minus 3. The kurtosis κ is defined to be the normalized fourth central moment - 3. In the second of these expressions, the hyphen used as a minus sign could be mistaken for a dash, so it's like saying: The kurtosis κ is defined to be the US president - George W. Bush.
Notice that in this context the usual skewness is not well defined, as for < the distribution does not admit 2nd or higher moments, and the usual skewness definition is the 3rd central moment. The reason this gives a stable distribution is that the characteristic function for the sum of two independent random variables equals the product of the ...