Probability distribution vs. probability function Is there a difference between the expressions "probability distribution" and "probability function" or are they just synonyms?
 A: "Probability distribution" denotes the law governing some random process. As noted in the comments, "probability function" is vague. Proper terminology would be "probability mass function" (PMF), "probability distribution function" (PDF) or "cumulative distribution function" (CDF).
The numerous misnomers for "probability distribution" may relate to the fact that the distribution of a random variable is characterized by the PMF, PDF, CDF, characteristic function, etc.
A: In my opinion there has been a general pedagogical failing with regard to defining the term "probability distribution". So I like this question every time it is asked. The term "probability distribution" is often used loosely in a way that causes genuine confusion, including for myself.
The term "probability function" is not so common, but I have seen it used as a synonym for "probability measure", which is a totally unambiguous term. 
The distribution of a random variable $X$ is the probability measure $\mu$ on $\mathbb R$ defined by $\mu(A) = P(X \in A)$ (for any Borel measurable subset $A$ of $\mathbb R$). This definition of "the distribution of a random variable $X$" is very standard; for example it is used in Folland, and probably every other rigorous textbook that covers probability.
Unfortunately, the PMF of a discrete random variable $X$ is sometimes also called a probability distribution or "the distribution of $X$". Same goes for the PDF of a continuous random variable $X$. This seems to be common in machine learning, for example. If someone asks you, "what's the distribution of $X$?" they expect you to tell them either the PMF or PDF of $X$, whichever one it has (assuming $X$ is either continuous or discrete).
Sometimes the term "distribution of $X$" is used as a synonym for "cumulative distribution function" of $X$ (for example by Sheldon Ross).
Sometimes a probability measure is called a "probability distribution" (by Knuth in Concrete Mathematics, for example).
So I say again, there has been a genuine pedagogical failing with this term. 
