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D: Laplace distribution, also known as the double exponential distribution, red curve (two straight lines in the log-scale plot), excess kurtosis = 3; S: hyperbolic secant distribution, orange curve, excess kurtosis = 2; L: logistic distribution, green curve, excess kurtosis = 1.2; N: normal distribution, black curve (inverted parabola in the ...
3.4 Kurtosis. 3.5 Meixner distribution. 3.6 ... the hyperbolic secant distribution is a continuous probability distribution whose probability density function and ...
If the function is a probability distribution, then the first moment is the expected value, the second central moment is the variance, the third standardized moment is the skewness, and the fourth standardized moment is the kurtosis. For a distribution of mass or probability on a bounded interval, the collection of all the moments (of all ...
The second is the traditional kurtosis, or fourth standardized moment: β 2 = γ 2 + 3. (Modern treatments define kurtosis γ 2 in terms of cumulants instead of moments, so that for a normal distribution we have γ 2 = 0 and β 2 = 3. Here we follow the historical precedent and use β 2.)
A fat-tailed distribution is a probability distribution that exhibits a large skewness or kurtosis, relative to that of either a normal distribution or an exponential distribution. [when defined as?] In common usage, the terms fat-tailed and heavy-tailed are sometimes synonymous; fat-tailed is sometimes also defined as a subset of heavy-tailed ...
A Bayesian account can be found in Gelman et al. [15] The degrees of freedom parameter controls the kurtosis of the distribution and is correlated with the scale parameter. The likelihood can have multiple local maxima and, as such, it is often necessary to fix the degrees of freedom at a fairly low value and estimate the other parameters ...
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Since , the excess kurtosis is always positive so the distribution is leptokurtic. [ 3 ] : 69 In other words, the tail of a geometric distribution decays faster than a Gaussian. [ 10 ] : 217