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A product distribution is a probability distribution constructed as the distribution of the product of random variables having two other known distributions. Given two statistically independent random variables X and Y, the distribution of the random variable Z that is formed as the product = is a product distribution.
The product of two Gaussian functions is a Gaussian, and the convolution of two Gaussian functions is also a Gaussian, with variance being the sum of the original variances: = +. The product of two Gaussian probability density functions (PDFs), though, is not in general a Gaussian PDF.
The squared Mahalanobis distance () is decomposed into a sum of k terms, each term being a product of three meaningful components. [6] Note that in the case when k = 1 {\displaystyle k=1} , the distribution reduces to a univariate normal distribution and the Mahalanobis distance reduces to the absolute value of the standard score .
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In probability theory, a probability density function (PDF), density function, or density of an absolutely 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 ...
Note that the above equation has three solutions, one at zero and two more with the opposite sign. By substituting the above equation, to the partial derivative of the log-likelihood w.r.t σ 2 {\displaystyle \sigma ^{2}} and equating it to zero, we get the following expression for the variance
Generalized hypergeometric functions include the (Gaussian) hypergeometric function and the confluent hypergeometric function as special cases, which in turn have many particular special functions as special cases, such as elementary functions, Bessel functions, and the classical orthogonal polynomials.
In the previous two integrals, n!! is the double factorial: for even n it is equal to the product of all even numbers from 2 to n, and for odd n it is the product of all odd numbers from 1 to n; additionally it is assumed that 0!! = (−1)!! = 1.