# What is the difference between $\mathbf P()$ and $p()$?

I am trying out the first probability problem at this link which looks something like this: I would like to ask what is the difference between $$\mathbf P(...)$$ and $$p_{...}(...)$$? Aren't they both just probability? Why bother using different symbols? Thanks.

• I think $p()$ is the joint density, whereas $\mathbf P()$ refers only to $Y$. Is just to distinguish them as we use $Y$ for the r.v. and $y$ for the particular realization. – Patricio Feb 14 at 8:39
• In this case $p_{X,Y}(x,y) = \mathbf P(X=x, Y=y)$ so you might want to say they are equivalent. Here $p_{X,Y}(x,y)$ takes the form of a function of $x$ and $y$ while $\mathbf P(X=x, Y=y)$ takes the form of the probability of an event, and you might want to draw that distinction. – Henry Feb 14 at 8:39

The probability of an event A is written as $$P(A)$$, $$p(A)$$, or $$Pr(A)$$.

$$\mathsf P(X=x)$$ is the probability for the event of $$X=x$$.
$$p_{\small X\!}(x)$$ is the probability mass function of random variable $$X$$, measured at $$x$$.
[Also $$p_{\small X\!}(x)$$ takes up slightly less typesetting on the page than $$\mathsf P(X=x)$$ , so can save space in long formulae.]