Averie Ferguson

2022-04-03

Does the logistic function uniquely satisfy these three conditions?

Given

$r\left(t\right)=\frac{f\left(t\right)}{1-F\left(t\right)}$ {Eq. 1}

where

$f\left(t\right)=\frac{dF}{dt}$ {Eq. 2}

and the conditions:

$\underset{t\to \mathrm{\infty}}{lim}r\left(t\right)=1$ {Eq. 3}

$\underset{t\to \mathrm{\infty}}{lim}F\left(t\right)=1$ {Eq. 4}

$\underset{t\to \mathrm{\infty}}{lim}f\left(t\right)=1-F\left(t\right)$ {Eq. 5}

I can think of just one function F satisfying these three conditions--the logistic function:

$F\left(t\right)=\frac{1}{1+{e}^{-t}}$ {Eq. 6}

(which can also be expressed$F\left(t\right)=r\left(t\right)$ )

Is this is the only function satisfying these conditions? If so, is there a way to prove it?

Given

where

and the conditions:

I can think of just one function F satisfying these three conditions--the logistic function:

(which can also be expressed

Is this is the only function satisfying these conditions? If so, is there a way to prove it?

Boehm98wy

Beginner2022-04-04Added 18 answers

So you can put as many things in the role of F as will fit in the role of r, i.e. things satisfying your Eq. 3.

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