Notation and meaning of a general probability distribution

I had two questions regarding the meaning and notation of a probability distribution when it is not specifically specified. For example, some papers jump right into their approach with notation like (page 2):

The goal of two-sample tests is to assess whether two samples, denoted by $$S_P$$$$P$$ and $$S_Q$$$$Q$$, are drawn from the same distribution.

Here is another example (page 1):

Given two sets of independent samples from unknown distributions $$P$$ and $$Q$$, a two-sample test decides whether to reject the null hypothesis that $$P = Q$$.

And a third example with slightly different notation (page 2):

Given a sequence of $$d$$-dimensional observations $$\{ x_1, . . . , x_t, . . .\}$$, with $$x_i ∈ \mathbb{R}^d$$, our goal is to detect the existence of a change-point such that before the change-point, samples are i.i.d from a distribution $$\mathbb{P}$$, while after the change-point, samples are i.i.d from a different distribution $$\mathbb{Q}$$.

1. In this context are $$P$$ (and $$Q$$) supposed to be CDFs? PDFs?
2. What's the typical notation for these situations just an italic capital letter or blackboard bold like $$\mathbb{P}$$ (and $$\mathbb{Q}$$)?

It seems like this tends to come up when describing unknown probability distributions.

• A CDF or PDF is one particular way to specify or describe a distribution. A description is any mathematical object that gives enough information to create a random variable with that distribution. There's no such thing as "typical notation" because it varies among communities, but what is common to them all are the concepts of random variable and distribution.
– whuber
Commented Jan 12, 2020 at 14:19
• @whuber So in this context, the notation $P$ and $Q$ is simply to differentiate between the two samples and is not specifically indicative of a CDF or PDF?
– guy
Commented Jan 12, 2020 at 16:53
• If I'm reading your quotations correctly, $P$ and $Q$ name two distributions that are used to model two samples. In particular, they do not refer to the (empirical) distributions of those samples, nor do they refer directly to those samples.
– whuber
Commented Jan 12, 2020 at 16:58
• @whuber That makes sense. Thanks. In that case, I will not use the blackboard bold notation even though a few papers use it as usually this notation refers to special sets like the set of rational numbers, the set of real numbers, etc.
– guy
Commented Jan 12, 2020 at 17:05
• BB bold also commonly refers to vector spaces, to the expectation operator, to a probability function, and many other things. Used consistently, it is effective in representing any category of mathematical objects one might like.
– whuber
Commented Jan 12, 2020 at 18:03

A "probability distribution" can be uniquely described either by its CDF or the corresponding probability measure. Contrarily, a density function is often not a unique description of a probability distribution, and so it would not usually be used for statements of the equality of distributions.$$^\dagger$$ Unfortunately, there is no standard notation for these kinds of problems, and usage differs substantially over different fields, so you really just have to read the statements in their context, to ensure you are interpreting the author correctly. The problem is compounded by the fact that an assertion of equivalence of distributions is already so clearly specified that some authors will play a bit "fast and loose" with the notation, and may actually use notation that is not strictly correct, but you are expected to understand what they intend anyway. (For example, you might occasionally see authors use the same notation for the "distribution" and also the density function.)
When using upper case letters like $$P$$ and $$Q$$ this is often (but not always) a reference to the CDF, whereas the blackboard-bold letters $$\mathbb{P}$$ and $$\mathbb{Q}$$ are often (but not always) a reference to the probability measure. In the absence of some contextual cues to the contrary, I would usually interpret them this way, but since there is no standard notation for these kinds of statements, you really just have to read the full paper, and you may even have to take an educated guess as to exactly what the author is referring to in the notation. Good luck!
$$^\dagger$$ For continuous random variables (dominated by Lebesgue measure), the density function can be altered at any countable set of points without changing the distribution. For discrete random variables (dominated by counting measure), the mass function is a unique representation of the distribution.
• Thanks for summarizing what @whuber and I discussed in the above comments. This indeed seems to be the situation I am in above. I am gonna interpret the ones in this case as probability measures even though i don't think it makes a huge difference at the end of the day. As long as someone understands that $P$ and $Q$ are unique ways to describe two samples in some probabilistic way and we must determine if they are identical or not.