Iv'e seen the use of both upper case and lower case letters for probability functions (e.g. $P$ vs $p$ or $Q$ vs $q$). I used to think these refer to discrete vs. continuous distributions, receptively. But then sometimes it seems that the distinction is between actual probabilities of events, (i.e. the actual scalar value of $P(A)$ or $P(X=x)$) and the functional associated with a random variable (like $p(X)$)
Is there a convention of using upper and lower case letter functions? If so, what is it?
It's up to you on how to use symbols with big or small letters. But there are some basic "assumptions" that you should have in mind: Use capital P for probabilities of events:If $A$ is an event,then $P(A)$ is the probability of $A$ happening. For functions, you can use small letters. For example,if $X$ is a random variable and $f$ is a function,then $f(X)$ is also a random variable. But again,no one would reject using a small letter for a probability. For instance you san say that $p = P(A)$ .