Conditioning on vs. fixing a random variable I am confused by the following notation, seen used by a professor in a course I'm taking.  $p(X|Y)$ denotes the conditioning of a distribution over a random variable $X$ by $Y$ and $p(X;y)$ denotes the distribution over $X$ where some variable $y$ is fixed.  I haven't been able to find any references about the latter notation online, but ostensibly, it is implied that $y$ is not a random variable.  Is this correct?  Otherwise, if $y$ were a random variable, the two notations would be equivalent, correct?
EDIT:
Note that I did not mean that $y$ is a value taken by $Y$.  My choice of variable names was poor, but please treat $y$ as a completely separate variable from $Y$, where $y$ may or may not be a random variable.  
 A: In probability theory generally, and as is the case here, upper case letters are used to denote random variables, and lower case letters are used to denote a specific (numerical, i.e., fixed) value taken on by the random variable.  Hence, P(X=x) means the probability that the random variable X has the value x.
So in your question, y denotes a specific, i.e., fixed, value taken on by the random variable Y.
If you don't like this convention, and think that it is too easy to confuse upper and lower case letters, don't blame me, I did not create it.
A: I've always understood the semicolon as delimiting random variables from distribution parameters. For example, the normal distribution is parameterized by mean and variance. I think this notation just allows you to talk about the parameters as variables, and discuss or alter their impact on the distribution of the random variables in the expression.
A: In the situations that I've seen, $p(X; y)$ is used in cases when $y$ is not a random variable, but some other parameter (constant, variable, etc., doesn't matter) on which you don't define a distribution.
You may also use the notation $p_y(X)$ instead of $p(X; y)$.  As a concrete example: Consider a distribution $p(X=x;y) \propto \exp(-y\; x^2)$, where $y$ is some constant parameter.
It was confusing at first, but you will get used to it.
