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In statistics, Bessel's correction is the use of n − 1 instead of n in the formula for the sample variance and sample standard deviation, [1] where n is the number of observations in a sample. This method corrects the bias in the estimation of the population variance.
Bessel functions describe the radial part of vibrations of a circular membrane.. Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions y(x) of Bessel's differential equation + + = for an arbitrary complex number, which represents the order of the Bessel function.
Correction factor versus sample size n.. When the random variable is normally distributed, a minor correction exists to eliminate the bias.To derive the correction, note that for normally distributed X, Cochran's theorem implies that () / has a chi square distribution with degrees of freedom and thus its square root, / has a chi distribution with degrees of freedom.
The Lerche–Newberger, or Newberger, sum rule, discovered by B. S. Newberger in 1982, [1] [2] [3] finds the sum of certain infinite series involving Bessel functions J α of the first kind. It states that if μ is any non-integer complex number, (,], and Re(α + β) > −1, then
In mathematics, a Jackson q-Bessel function (or basic Bessel function) is one of the three q-analogs of the Bessel function introduced by Jackson (1906a, 1906b, 1905a, 1905b). The third Jackson q -Bessel function is the same as the Hahn–Exton q -Bessel function .
The reason for the factor n − 1 rather than n is essentially the same as the reason for the same factor appearing in unbiased estimates of sample variances and sample covariances, which relates to the fact that the mean is not known and is replaced by the sample mean (see Bessel's correction).
Bessel beam; Bessel ellipsoid; Bessel function in mathematics; Bessel's inequality in mathematics; Bessel's correction in statistics; Bessel filter, a linear filter often used in audio crossover systems; Bessel transform, also known as Fourier-Bessel transform or Hankel transform; Bessel window, in signal processing
In mathematics, the Hahn–Exton q-Bessel function or the third Jackson q-Bessel function is a q-analog of the Bessel function, and satisfies the Hahn-Exton q-difference equation (Swarttouw ). This function was introduced by Hahn ( 1953 ) in a special case and by Exton ( 1983 ) in general.