Interpretation of joint pdf/product of marginal pdfs If we have two Random Variables X and Y, what is the interpretation for the three cases:
a] p(X=x,Y=y) = $p_1$(X=x) * $p_2$(Y=y)
b] p(X=x,Y=y) < $p_1$(X=x) * $p_2$(Y=y)
c] p(X=x,Y=y) > $p_1$(X=x) * $p_2$(Y=y)  
where p -> joint pdf of X and Y,
$p_1$ -> marginal pdf of X
$p_2$ -> marginal pdf of Y
 A: I assume that this statement applies for specific values of $x,y$ because I'm not sure that (b) or (c) is possible for all $x,y$. Also, the way you've written it here these must be discrete random variables (e.g. by use of the '=') so these are actually pmfs, not pdfs. That being said, first note that $P(X = x, Y = y) = P(X = x | Y = y) P(Y = y)$ (and similarly with the roles of $X$ and $Y$ reversed). So: 
(a) implies that $P(X = x | Y = y) = P(X = x)$, which is the definition of independence. That is, knowing that $Y = y$ tells us nothing about the event $X = x$. 
(b) implies that $P(X = x | Y = y) < P(X = x)$, which tells us that knowledge that $Y=y$ decreases the probability that $X=x$, meaning that the events $X=x$ and $Y=y$ are negatively ''associated'' with each other (I do not use the term correlation here because that is a statement about two random variables, but the problem stated here is one about probabilities of specific events). 
Similarly, (c) gives an indication that $X=x$ and $Y=y$ are two events that are positively associated in some sense. 
A: Two random variables $X$ and $Y$ are independent if $P(X=x, Y=y) = P(X=x)P(Y=y)$, i.e. their joint pdf factorizes. E.g. $X$ = getting heads on the first toss of a coin, $Y$ = getting heads on the second toss of the coin. 
In general, for random variables $X$ and $Y$, $P(X = x, Y = y) \leq P(X=x)$ and $P(X = x, Y=y) \leq P(Y=y)$. This implies that (c) can't hold true. There's no specific interpretation to (b).
A: Since I can't make comments (haven't got enough points yet), I writing them in the form of answer. The word "pdf" stands for "probability density function", not for "probability distribution function". In your particular case, of course, you mean some discrete distributions, which are referred to as "pmf" (probability mass function). So, please do use the correct terminology. 
