Two cards are drawn at random without replacement from an ordinary deck. Let X be the number of hearts and Y the number of black cards obtained.
I know the joint pdf is f(x,y) = $f(x,y) = \frac{ \bigl( \begin{smallmatrix} 13 \\ x \end{smallmatrix} \bigr) \bigl( \begin{smallmatrix} 26\\ y\end{smallmatrix} \bigr) \bigl( \begin{smallmatrix} 13\\ 2-x-y \end{smallmatrix} \bigr)}{ \bigl( \begin{smallmatrix} 52 \\ 2 \end{smallmatrix} \bigr)}$
with marginal pdf's of $f(x) = \frac{\bigl( \begin{smallmatrix} 13 \\ x \end{smallmatrix} \bigr) \bigl( \begin{smallmatrix} 39 \\ 2-x \end{smallmatrix} \bigr) }{ \bigl( \begin{smallmatrix} 52 \\ 2 \end{smallmatrix} \bigr)}$ and $f(y) = \frac{\bigl( \begin{smallmatrix} 26 \\ y \end{smallmatrix} \bigr) \bigl( \begin{smallmatrix} 26 \\ 2-y \end{smallmatrix} \bigr) }{ \bigl( \begin{smallmatrix} 52 \\ 2 \end{smallmatrix} \bigr)}$.
Are X and Y independent?
No they arent, if they were independent then we know that $f_{XY}= f_x f_y$
This is not the case here
In fact the value of X and Y are dependent because looking at the pdf, we can see that if $X>1$, then $Y<1$ as well