# 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

## Related Questions

### Total expectation under measure theory

1  Asked on December 6, 2020 by jiexiong687691

### complex norm inequality

1  Asked on December 6, 2020 by stranger

### How do you define “independence” in combinatorics?

3  Asked on December 6, 2020 by trivial-math-is-difficult

### Show that the function $f(z)=frac{1-cos z}{z^n}$ has no anti-derivative on $mathbb Csetminus{0}$ if $n$ is an odd integer.

1  Asked on December 6, 2020 by riyasudheen-t-k

### Checking a ring is not Cohen-Macaulay

2  Asked on December 6, 2020

### Unique representation: $a!cdot b!cdot c!=m!cdot n!cdot p!$

4  Asked on December 6, 2020 by felipeuni

0  Asked on December 6, 2020 by twosigma

### Is $x^2$ analytic in $mathbb{R}$

1  Asked on December 5, 2020 by joey

### The best strategy to maximize the last number on a die.

2  Asked on December 5, 2020 by student

### How to find bounds when doing a double integral?

1  Asked on December 5, 2020 by dr-suess-official

### A simple looking second-order DE is giving me tough time

1  Asked on December 5, 2020 by curious-2-learn

### Convergence of a series of integrals: $S=sum_{ngeq 1}(-1)^{n}int_{n}^{n+1}frac{1}{t e^t},dt$

3  Asked on December 5, 2020 by ramana

### Understanding Seifert Van Kampen

0  Asked on December 5, 2020 by moooose

### Trying to evaluate a complex integral?

4  Asked on December 5, 2020 by gray

### Random variable $X$ has uniform distribution on section $[0,2]$. What’s the expected value of variable $Y=frac{X^{4}}{2}$

2  Asked on December 5, 2020 by fakeraker-p

### Show that this iterative Richardson iteration may diverge

1  Asked on December 5, 2020

### Estimating number of infected people and getting bounds of its probability based on a few samples.

1  Asked on December 5, 2020 by user_9