TITLE: Absolute moment existence
QUESTION [2 upvotes]: I have given
$$F_X(x) = \frac{1}{1+e^{-x}} ; \quad x \geq0$$
I have to calculate $k$, s.t. $E[|X|^k]<\infty$
I did calculate the density by $f_X(x) = \frac{dF}{dx} =\frac{e^{-x}}{(1+e^{-x})^2} $ . Using this to calculate the absolute moments, I found it to be very cumbersome so I think that this might not be the best way to solve this. Does anyone have thoughts?
Best

REPLY [2 votes]: Your density is not correct.  Differentiation of the CDF gives $$f_X(x) = \frac{dF_X}{dx} = -(1 + e^{-x})^{-2} (-e^{-x}) = \frac{e^{-x}}{(1+e^{-x})^2} = \frac{e^{-x}}{e^{-x}(e^{x/2} + e^{-x/2})^2} = \frac{1}{4 \cosh^2 \frac{x}{2}}.$$
If $k$ is a positive integer, it is not difficult to show $\operatorname{E}[|X|^k] < \infty$.  I leave this as an exercise; in particular, one might wish to consider the moment-generating function of $|X|$.
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