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Show that the density is $f(x)=frac{e^{-x}}{left(1+e^{-x}right)^{2}}$

A positive random variable $X$ has the logistic distribution if its distribution function is given by
$$
F(x)=P(X leq x)=frac{1}{1+e^{-(x-mu) / beta}} ; quad(x>0)
$$

for parameters $(mu, beta)$, $beta>0$.
a) Show that if $mu=0$ and $beta=1$, then a density for $X$ is given by
$$
f(x)=frac{e^{-x}}{left(1+e^{-x}right)^{2}}
$$


I’m doing a course in measure theory, so I do not know if it is enough to differentiates (I doubt it is).
So what will a more rigorous approach be?

Also, I think there is some typos in the problem. $F$ and $f$ should have been $F_X$ and $f_X$.

Mathematics Asked on December 30, 2020

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