I have been reading a paper for a while, but still get confused when the author uses the term 'law' and the term 'distribution'.
So, he literally says this: "denote by $P$ the joint distribution of $X$ and $Y$, thus if we factorize the marginal law $P$ on $Y$, as..."
and another one: "... the density of this law ... " ?? what does this mean?
So, is it that Law and distribution refer to the same? I mean a PDF?
Are there any other terms I should be familiar with so, I would be easier for me to read papers in these topics? Please, feel free to type any terms you may have found while doing your research.
I have seen that people use 'density', 'distribution'.
I often see the words "distribution" and "law" as generic terms for a description of a random variable. This can be made concrete as a CDF or (sometimes) as a PDF. Thus the law/distribution is a more abstract thing attached to the random variable whereas the CDF or PDF is a concrete representation of the information in the distribution. This is probably the sense the person meant it when they said "the density of this law."
I also often see the word "law" used a lot in a technical sense as another type of mathematical object that characterizes a distribution. Namely, the law is the probability measure induced by the random variable. For a real-valued random variable, we have a probability space $(\mathbb R, \mathcal B(\mathbb R), \mathbb P_X)$ where $\mathcal B(\mathbb R)$ is the Borel sigma algebra on $\mathbb R$ and the law $\mathbb P_X$ is the measure such that for $A\in \mathcal B(\mathbb R),$ $$\mathbb P_X(A) = \mathbb P(X\in A).$$ It can be defined as a push-forward of the original probability space $(\Omega,\Sigma,\mathbb P)$ on which the random variable $X:\Omega\to \mathbb R$ is defined. In other words $ \mathbb P_X(A) = \mathbb P(X^{-1}(A)).$