Determine if X and Y are independent based on joint pdf

Question

Suppose $X$ takes on the values $1,2,...,n$ while $Y$ takes on values $1,2,3,...,m$. The joint pdf is $1/(nm)$ for all possible pairs of values. Are $X$ and $Y$ independent? I know in order to determine if they are independent for this, I need to find the marginals for both $x$ and $y$, and multiply them together to see if it equals the joint pdf. So I got $f_x(x) = \ln(m)/n$ and $f_y(y) = \ln(n)/m$. This is obviously not equal to $1/(nm)$, but the answer says they are independent. How? Am I integrating wrong?

Answer 1

This is a discrete random variable.

for $x \in \{ 1, \ldots, n\}$ $$P_X(X=x) = \sum_{y \in \{ 1, \ldots m\}} P(X=x,Y=y)=\frac{m}{mn}=\frac1n$$

for $y \in \{ 1, \ldots, m\}$ $$P_Y(Y=y) = \sum_{x \in \{ 1, \ldots n\}} P(X=x,Y=y)=\frac{n}{mn}=\frac1m$$