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The logistic distribution is well known. For example, the standard pdf of the logistic distribution is given as: $$ f_X(x) = \frac{e^x}{(1+e^x)^2},\,\,-\infty\lt x\lt \infty~~~~~~~~~~(1)$$ My question is this: How did this distribution come about? How can one derive it?

I would also be grateful if I could get a reference where I could find the derivation of some of the well-known probability distributions.

Gorg
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    What [b]definition[/b] of "logistic distribution" are you using? – user247327 Sep 01 '16 at 13:39
  • If $x$ is replaced by $-x$ your PDF is just the derivative of the CDF $F(x)=\frac{1}{1+e^{-x}}$ which is a special case of the https://en.wikipedia.org/wiki/Logistic_function – gammatester Sep 01 '16 at 13:54
  • It tells you at Wikipedia, it is similar to the normal distribution, the PDF can be integrated analytically and the PDF has heavier tails, which might be wanted for some applications. It was probably discovered from a mathematical model, or just spotted as a candidate for a likely distribution. – Cato Sep 01 '16 at 13:58
  • @user247327 How many definitions are there? – Gorg Sep 01 '16 at 13:59

1 Answers1

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Ultimately it stems from turning a probability $p$ defined on the interval $(0,1)$ into the logarithm of its odds: $$g(p)=\log_e\left(\dfrac{p}{1-p}\right)$$ taking values in the interval $(-\infty,+\infty)$ and called the logit function.

So $\exp(g(p))=\dfrac{p}{1-p}$ and thus $p=\dfrac{\exp(g(p))}{1+\exp(g(p))}$ making the inverse function $$g^{-1}(x)=\dfrac{\exp(x)}{1+\exp(x)}$$ as a continuous bijective increasing function $(-\infty,+\infty) \to (0,1)$ and called the logistic function. You can therefore use this as the cumulative distribution function of a random variable and taking its derivative gives a density function of $$f(x)=\dfrac{\exp(x)}{(1+\exp(x))^2}$$

Henry
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    Thanks for your answer. Is there a rationale behind considering the logarithm of odds? – Gorg Sep 01 '16 at 21:15
  • @Gorg: If you are doing things like linear regression, then being able to work with something which has the range $(-\infty,+\infty)$ helps avoid predicting probabilities (or even just fractions of a whole) less than $0$ or greater than $1$. You may also want a measure which says going from $0.1%$ to $0.2%$ is roughly as substantial a change as going from $1%$ to $2%$ (pointing at logarithms) and as going from $99.8%$ to $99.9%$ (pointing at something symmetric), and log-odds does this neatly. – Henry Sep 02 '16 at 07:28