What is the probability distribution used in logistic regression called? In logistic regression, we set the probability of predicting a target $y$ given a data $x$ as,
$\Pr(Y = 1|X;w) = \dfrac{\exp(w^TX)}{(1+\exp(w^TX))}$
What is exactly this probability distribution (or more accurately, conditional probability mass function) called? 
I tried to look up logistic distribution but it doesn't look the same https://en.wikipedia.org/wiki/Logistic_distribution
 A: Since $Y_i$ is a binary variable, its distribution is the Bernoulli distribution:
$$Y_i | \mathbf{x}, \mathbf{w} \sim \text{Bern} \Bigg( \text{Prob} = \frac{\exp(\mathbf{w}^\text{T} \mathbf{x})}{1 + \exp(\mathbf{w}^\text{T} \mathbf{x})} \Bigg).$$
One alternative way of looking at the logistic regression is to regard the observed response variable as a discretisation of an underlying "latent variable", where the latter has a logistic distribution.  In this (equivalent) alternative formulation, we have an observed response variable $Y_i \equiv \mathbb{I}(\tilde{Y}_i > 0)$, with the underlying latent response having the distribution:
$$\tilde{Y}_i | \mathbf{x}, \mathbf{w} \sim \text{Logistic} \Bigg( \text{Location} = \mathbf{w}^\text{T} \mathbf{x}, \ \text{Scale} = 1 \Bigg).$$
A: In your link, you have the cumulative distribution function for the logistic distribution as $$\frac{1}{1+e^{-\frac{x-\mu}{s}}}$$
while in your question you have $$\dfrac{\exp(w^TX)}{(1+\exp(w^TX))} \text{ which is } \dfrac{1}{1+e^{-w^TX}}$$ 
and these are essentially the same so long as $w^TX$ has mean $0$ and variance $1$.  
You can see that the first expression is a cumulative distribution function and this expression approaches $0$ when $x$ is very negative but approaches $1$ when $x$ is large and positive.  This is what want from your logistic regression: the predicted conditional probability of a positive result $(Y=1)$ increases towards $1$ as $w^TX$ increases     
