This could be a stupid question but, since sigmoid function maps values between $-\infty$ and $\infty$ to values between 0 and 1, I thought it could be a probability distribution. However when I take the integral, I get
$$ x + \ln(1 + e^{-x}) $$
Little cleanup
$$ \ln e^{x} + \ln(1+e^{-x}) $$
$$ \ln (e^{x}\cdot 1 + e^{x}e^{-x}) $$
$$ \ln (1 + e^{x} ) $$
When I evaluate this for bounds $-\infty$ and $\infty$, I do not get the result 1. Did I do something wrong, and is there another form of the sigmoid function (perhaps normalized) to use it as a probability distribution?
Best Answer
Sigmoid
$$f(x) = 1 / (1 + e^{-x})$$
is not a probability distribution function (it approaches 0 on -∞ and 1 on +∞, so its integral will also be ∞). However it is a cumulative distribution function of logistic distribution, if thats what you mean.