We know that for two discrete random variables $X$ and $Y$ to be independent, the joint density of random vector $Z=(X,Y)$ has to be equal to product of marginal densities of $X$ and $Y$.
Now, this means:
a) To establish independence of $X$ and $Y$, we gotta test the above equality for every vector of euclidean plane.
b) To establish independence of $X$ and $Y$, we can test the above equality for only one vector of euclidean plane, and this suffices.
Thanks.
Best Answer
For discrete random variables taking on finite sets of values, the joint probability mass function can be viewed as a matrix $P_{X,Y}$ with, say $m$ rows and $n$ columns, while the marginal mass functions can be viewed as $1\times m$ and $1\times n$ matrices (row vectors) $P_X$ and $P_Y$. Independence requires you to check whether $P_{X,Y} = P_X^TP_Y$, and this requires you to verify this matrix equality (that is, check whether the entries are all the same on both sides). So, $m\times n$ tests.