Confused about the solutions to a joint PMF question

In this question, the l and k are "new." If these letters appear in previous solutions, the slate is wiped clean now and we're treating them like we haven't seen them before.

When we are calculating a pmf, we are looking for

p_{Y1, Y2}(a,b) = P(Y1 = a, Y2 = b)

where a and b are some numbers. (Here, I'm using a and b instead of l and k, but you can use pretty much any character you want.) But, as mentioned in the question, Y1 = X1 - 1 and Y2 = X1 -1 + X2 - 1.

So if Y1 is equal to a, then using the above equation X1 must be equal to a + 1. You can play a similar game with X2 to get  b - a + 1. So then,

p_{Y1, Y2}(a,b) = P(Y1 = a, Y2 = b)
= P(X1 = a+1, X2 = b - a + 1)
= p_{X1, X2}(a + 1, b - a + 1)

So you can see the joint pmf of Y1 and Y2 will be very very related to the joint pmf of X1 and X2, but not quite the same.

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