Which random variable has the characteristic function $f(t)=\frac{e^{it}}{1-it}$

Question

Which random variable has the characteristic function $$f(t)=\frac{e^{it}}{1-it}$$

This is quite important for me to know, I know I have seen it somewhere, but I cant remember which random variable.

Answer 1

The probability distribution $p(x)$ can be obtained via inverse Fourier-transformation $$p(x) = \int \frac{dt}{2\pi} e^{-i t x} \frac{e^{it}}{1-it}.$$ Performing the integral, we obtain the distribution $$p(x) = e^{1-x} $$ on $x\geq 1$.