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

0 Answers

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