Relation between logistic regression and logistic distribution

Question

When we are using logistic regression, we can get the probability that $y$ belongs to class $1$ as follows:

$P(y=1|x;\theta)=\frac{1}{1+\exp(-\theta^Tx)}$.

PDF of a logistic distribution is given as follows:

$f(x)=\frac{\exp(-x)}{(1+\exp(-x))^2}$

and this is a continuous distribution.

Here are my questions:

1. Conditional distribution of $y|x$ is logistic distribution. Is this right? Or is it a categorical distribution?
2. PDF at a single point is zero. So how can we get probabilities in logistic regression if it is a continuous distribution?
3. As a side question, what is it we get when we evaluate a PDF at a single point? For example, in scipy.stats, norm().pdf(3) returns a value, what is it?

Answer 1

1. Conditional distribution of | is logistic distribution. Is this right? Or is it a categorical distribution?

No, $y$ is a binary variable so cannot have a logistic distribution. The conditional distribution of $y\mid x$ is binomial.

2. PDF at a single point is zero. So how can we get probabilities in logistic regression if it is a continuous distribution?

$y$ is categorical, so does not have a pdf, it has a pmf (probability mass function.) By the way, the PDF at a single point does not need to be zero, see Can a probability distribution value exceeding 1 be OK? for intuition.

3. As a side question, what is it we get when we evaluate a PDF at a single point? For example, in scipy.stats, norm().pdf(3) returns a value, what is it?

See the answer above.

For the question in the title about relationship logistic distribution and logistic regression see How is Logistic Regression related to Logistic Distribution?