Smoothness (probability theory)
Introduction In probability theory and statistics, '''smoothness''' of a density function is a measure which determines how many times the density function can be differentiated, or equivalently the limiting behavior of distribution’s characteristic function.Formally, we call the distribution of a random variable ''X'' '''ordinary smooth''' of order ''β''  if its characteristic function satisfies: for some positive constants ''d''0, ''d''1, ''β''. The examples of such distributions are gamma, exponential, uniform, etc.The distribution is called '''supersmooth''' of order ''β''   if its characteristic function satisfies: for some positive constants ''d''0, ''d''1, ''β'', ''γ'' and constants ''β''0, ''β''1.Such supersmooth distributions have derivatives of all orders. Examples: normal, Cauchy, mixture normal.
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