The numbers are supposedly chosen from an infinite distribution...
There is no valid way to "sample" an infinite distribution of integers to get x and y, nor is there a way to get r from an infinite real distribution.
Sticking to the integers - the chance of "sampling" any given integer is zero. The chance of "sampling this distribution" with an integer between and two arbitrary integers, no matter how ridiculously large, is also zero.
Put another way, there is no way to define the probability of an integer "thus chosen" within any arbitrary range to be anything other than zero.
All values p(i) are zero.
Now, if we could put some constraints on the values the game show host can choose from, and the probabilities thereof, we could effectively integrate over ranges and talk about probabilities and formulate a winning strategy...