Solved – Find the pmf of a bivariate distribution when rolling a black and red four-sided die

Roll a pair of four-sided dice, one red and one black. Let $X$ equal the outcome on the red die and let $Y$ equal the sum of the two dice. Define the joint pmf on the space.

So far I have $X = 1,2,3,4$ and $Y = 2,3,4,5,6,7,8$. Each outcome on each die has a $frac{1}{4}$ probability of being rolled and thus each outcome of the combined rolls is $frac{1}{16}$. There are two ways to make each value in $Y$. For example to make $3$ we could have $2$ on the red die and $1$ on the black or $1$ on the red die and $2$ on the black die. $frac{1}{16} + frac{1}{16} = frac{2}{16}$. Therefore the pmf should = $frac{2}{16}$. This is not the answer, however :). Ideas?

Writing this up led me to the solution.

When we plug in values into the pmf $f(x,y)$ such as $f(1,2)$ we must recall that we have already decided what the value of $X$ is. Although there are two ways to sum two four-sided to get the value $2$, we have already set $X=1$ and therefore there is only one value ($1$) that the remaining die may have to make $2$. Thus there is only $1$ way to create $2$ given the parameters of the pmf and thus the probability for this and all rolls is static.

$f(x,y) = frac{1}{16}$


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