00:01
In this question we have a geometric random variable y.
00:04
So y is geometric with a parameter p, and that means the probability y equals y is 1 minus p to the y minus 1 times p.
00:14
Now we want to find the expected value of y times y minus 1.
00:23
So that's the expected value of y, so that's going to be the sum, sorry, from y equals 1 to infinity, y, y minus 1, 1 minus p to the y minus 1 times p.
00:39
But that's just p over 1 minus p, sorry, it's p times 1 minus p times the sum from y equals 1 to infinity, y, y minus 1 times 1 minus p to the y minus 2.
01:01
But actually because when y equals 1, the y minus 1 term will make it 0, we can actually sum from y equals 2.
01:12
So we're going to be summing from y equals 2.
01:17
But actually because this is being nullified for 0 and 1, we're actually summing from y equals 0, so we're including all these terms.
01:26
But we've brought down a factor of y, y minus 1, so this is p 1 minus p times d 2 by d 1 minus p squared of the sum from y equals 0 to infinity, 1 minus p to the y.
01:43
So let's find this sum...