Editing Kurtosis (section)

=== Other well-known distributions ===
[[Image:Standard symmetric pdfs.svg|300px|thumb|[[Probability density function]]s for selected distributions with [[expected value|mean]] 0, [[variance]] 1 and different excess kurtosis]]
[[Image:Standard symmetric pdfs logscale.svg|300px|thumb|[[Logarithm]]s of [[probability density function]]s for selected distributions with [[expected value|mean]] 0, [[variance]] 1 and different excess kurtosis]]

Several well-known, unimodal, and symmetric distributions from different parametric families are compared here.  Each has a mean and skewness of zero. The parameters have been chosen to result in a variance equal to 1 in each case. The images on the right show curves for the following seven densities, on a [[linear scale]] and [[logarithmic scale]]:

* 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 log-scale plot), excess kurtosis = 0
* C: [[raised cosine distribution]], cyan curve, excess kurtosis = −0.593762...
* W: [[Wigner semicircle distribution]], blue curve, excess kurtosis = −1
* U: [[uniform distribution (continuous)|uniform distribution]], magenta curve (shown for clarity as a rectangle in both images), excess kurtosis = −1.2.

Note that in these cases the platykurtic densities have bounded [[Support (mathematics)|support]], whereas the densities with positive or zero excess kurtosis are supported on the whole [[real line]].

One cannot infer that high or low kurtosis distributions have the characteristics indicated by these examples. There exist platykurtic densities with infinite support,
*e.g., [[exponential power distribution]]s with sufficiently large shape parameter ''b''

and there exist leptokurtic densities with finite support.
*e.g., a distribution that is uniform between −3 and −0.3, between −0.3 and 0.3, and between 0.3 and 3, with the same density in the (−3, −0.3) and (0.3, 3) intervals, but with 20 times more density in the (−0.3, 0.3) interval

Also, there exist platykurtic densities with infinite peakedness,
*e.g., an equal mixture of the [[beta distribution]] with parameters 0.5 and 1 with its reflection about 0.0

and there exist leptokurtic densities that appear flat-topped,
*e.g., a mixture of distribution that is uniform between −1 and 1 with a T(4.0000001) [[Student's t-distribution]], with mixing probabilities 0.999 and 0.001.

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