Why are the rows of the table linearly dependent for two independent RVs?

Question

I don't understand why the rows of this joint-distribution table are dependent for two, independent RVs.

What's the intuition here?

Answer 1

The key is what Math1000 says in the comments. We have $f_{XY}(x,y) = f_X(x)f_Y(y).$ A row of the table is for constant $x,$ so we have $$ f_{XY}(0, y) = f_X(0) f_Y(y)\\ f_{XY}(1, y) = f_X(1) f_Y(y)$$ etc, so for instance $$ f_{XY}(0,y) = \frac{f_X(1)}{f_X(0)}f_{XY}(1,y).$$

Some lingering questions:

1. What if $f_X(i)=0$ for some or several $i$?
2. This only shows one direction. How do you show that if the rows of the table are proportional, then $X$ and $Y$ are independent?