The characteristic function of a random variable $X$ is $\phi(t)=e^{-\sqrt{|t|}}$, we now want to calculate the cumulative distribution function of $X$. It seems that it's hard to do the integration through inverse fourier transform $\dots$
2026-03-25 22:09:45.1774476585
given the characteristic function $e^{-\sqrt{|t|}}$ , how to calculate the cumulative distribution function？
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This is a symmetric stable distribution with index $\frac 1 2$. There is no simple expression for the density function but properties of this distribution have been studied well. There is also a series representation of the density function.