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I've found on wikipedia for Logistic function (http://en.wikipedia.org/wiki/Logistic_function) they have the formula for a Logistic curve:

$P(t) = 1 / (1 + e^{-t})$

and they have a diagram of the curve. What is the formula for the cumulative distribution function? I think the cdf curve would look like the logistic curve, but rise faster, but I don't know.


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This IS the cumulative distribution function, en.wikipedia.org/wiki/Logistic_function#In_statistics. –  Did Aug 13 '11 at 22:34
thank you, I didn't realize that –  d l Aug 13 '11 at 22:44
Here's a nice rule of thumb: PDFs tend to look like humps, while CDFs tend to be "sigmoidal" (S-shaped). The logistic function is sigmoidal, so... –  Ｊ. Ｍ. Aug 14 '11 at 0:20
The area under a density function is $1$. So the picture instantly tells you $P(t)$ is not a density. –  André Nicolas Aug 14 '11 at 1:43

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This function $P$ is the cumulative probability distribution function of a probability distribution, sometimes called the logistic distribution. However, the question as phrased, if taken literally, does not make sense: you're asking "What is the cdf for the distribution" before you've said which distribution you have in mind. It is not the case that whenever you write about the logistic function, you're talking about some probability distribution. For example, the logistic function is the solution to a differential equation that is a simple model for population growth with a carrying capacity.

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Thank you, this makes it clear, I missed the relationship at first, which seems to be: the logistic distribution is the probability density function, which has a hump in the middle and area underneath = 1; and the logistic function (referenced in my question) is the cumulative distribution function, and sigmoidal in shape. –  d l Aug 14 '11 at 5:37

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