Can a marginal p.m.f. ever be exactly equal to the joint p.m.f. Let $X$ & $Y$ be two random variables (discrete or continuous). Are there any simple examples where the joint p.m.f. (or p.d.f.) equals one of the marginal p.m.f.s of $X$ or $Y$? I am just looking for examples, if any exist. 
 A: Technically, no, since the joint distribution function is a function of two variables, while the (marginal) distribution function of $X$ is a function of one variable. 
However, if $Y=a$ with probability $1$, and we forget about the distribution function of $(X,Y)$ when $y\ne a$, since $y\ne a$ "can't happen," then we do get what feels like equality. It really isn't. 
A: I'd say the idea of a joint PMF/PDF "equaling" a marginal PMF/PDF isn't really the right way to think about things. In the case you mentioned, the joint PMF is a function of two variables wehereas the marginal PMFs are functions of one variable. Saying one is equal or unequal to the other doesn't really make sense.
With the joint PMF $f_{X,Y}(x,y)$, we care about the outcome $X=x$ and $Y=y$ happening together at the same time.
The marginal PMF $f_X(x)$  is the probability of the random variable $X$ taking on the value $X=x$, regardless of the outcome of the random variable $Y$. When we marginalize over $Y$, we are saying "we don't care about the value of $Y$ anymore", so we sum over the values of all possible $y$'s to get rid of them.
$f_X(x)=\sum_y f_{X,Y}(x,y)$
A: A joint p.m.f. is a function of two variables, and is zero with at most  countably many exceptions.  It's impossible to have, say, 
$P(X=x,Y=y) = P(X=x)$ when $P(Y=y) = 0$ and $P(X=x) \ne 0$.
Similarly, for a joint pdf it's impossible to have $f_{X,Y}(x,y) = f_X(x)$ for all $y$.  It is possible to have $f_{X,Y}(x,y) = f_X(x)$ for $y$ in an interval  $[a,a+1]$ such that $P(a \le Y \le a+1) = 1$.  Simply let $Y$  have uniform distribution on this interval, and $X$ any continuous distribution, with $X$ and $Y$ independent. 
