Let Y be a random variable which takes values on the integer

Let Y be a random variable which takes values on the integers. Show that for every integer k we have P(Y = k) = P(Y k) P(Y k 1).

Solution

Note that, as Y only takes on integers,

P(Y<=k) = P(Y=k) + P(Y = k-1) + P(Y = k-2) + P(Y = k-3) ...

P(Y<=k -1) = P(Y = k-1) + P(Y = k-2) + P(Y = k-3) +...

Thus, subtracting these two, we see that all terms cancel except P(Y = k),

P(Y = k) = P(Y<=k) - P(Y<=k-1) [DONE]