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Common Way to Write Joint CDF of Discrete Random Variables
I am wondering if there is any efficient way to write the joint CDF for two random variables I am working with.
The setup is as follows: Toss two fair six-sided dice. Suppose that one die is red and the other is green. Define $X$ as the sum of dots showing on the two dice. Define $Y$ as the number of dots showing on the red die minus the number of dots showing on the green die.
I can easily find the $CDF$ or $PDF$ for $X$ and $Y$. The book I am working out of has creating the joing $CDF$ of $X$ and $Y$ as a problem. Is there a simple way to do this? It seems to me that this would turn into listing a very large number of possibilities.