Of well-known distributions Kurtosis

probability density functions selected distributions mean 0, variance 1 , different excess kurtosis



logarithms of probability density functions selected distributions mean 0, variance 1 , different excess kurtosis

several well-known, unimodal , symmetric distributions different parametric families compared here. each has mean , skewness of zero. parameters have been chosen result in variance equal 1 in each case. images on right show curves following 7 densities, on linear scale , logarithmic scale:

d: laplace distribution, known double exponential distribution, red curve (two straight lines in 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 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, magenta curve (shown clarity rectangle in both images), excess kurtosis = −1.2.

note in these cases platykurtic densities have bounded support, whereas densities positive or 0 excess kurtosis supported on whole real line.

there exist platykurtic densities infinite support,

e.g., exponential power distributions sufficiently large shape parameter b

and there exist leptokurtic densities finite support.

e.g., distribution uniform between −3 , −0.3, between −0.3 , 0.3, , between 0.3 , 3, same density in (−3, −0.3) , (0.3, 3) intervals, 20 times more density in (−0.3, 0.3) interval