Stretched exponential function: Difference between revisions

Revision as of 00:23, 13 September 2020

Revision as of 14:55, 17 October 2020

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:<math>f_\beta (t) = e^{ -t^\beta }</math>

is obtained by inserting a fractional [[power law]] into the [[exponential function]].

In most applications, it is meaningful only for argumentsargument ''t'' between 0 and +∞. With ''β''&nbsp;=&nbsp;1, the usual exponential function is recovered. With a ''stretching exponent'' ''β'' between 0 and 1, the graph of log&nbsp;''f'' versus ''t'' is characteristically ''stretched'', hence the name of the function. The '''compressed exponential function''' (with ''β''&nbsp;>&nbsp;1) has less practical importance, with the notable exception of ''β''&nbsp;=&nbsp;2, which gives the [[normal distribution]].

In mathematics, the stretched exponential is also known as the [[Cumulative distribution function#Complementary cumulative distribution function (tail distribution)|complementary cumulative]] [[Weibull distribution]]. The stretched exponential is also the [[characteristic function (probability theory)|characteristic function]], basically the [[Fourier transform]], of the [[stable distribution|Lévy symmetric alpha-stable distribution]].