Your chances of living to be 1000 are approximately 1 in e, 37%, yes. But that has nothing to do with the average age people live to. I don't get 1000 as the average lifespan, though, I get 999. Can someone explain where I'm going wrong with this reasoning?
If there is a 0.001 chance of dying every year, the average lifespan will be 0(0.001) + 1(0.001)(0.999) + 2(0.001)(0.999)^2 + 3(0.001)(0.999)^3 + ... , or \sum_{i=0}^\infty (i(0.001)(0.999)^i). The idea is that to live to the age of 0, you have to die in your first year; to live to be three, you have to live through exactly three years and then die, and so on. Then, \sum_{i=0}^\infty ix^i is x / (1-x)^2 .
So the sum of interest is 0.001 times (0.999) / (0.000001), which is exactly 999.
If there is a 0.001 chance of dying every year, the average lifespan will be 0(0.001) + 1(0.001)(0.999) + 2(0.001)(0.999)^2 + 3(0.001)(0.999)^3 + ... , or \sum_{i=0}^\infty (i(0.001)(0.999)^i). The idea is that to live to the age of 0, you have to die in your first year; to live to be three, you have to live through exactly three years and then die, and so on. Then, \sum_{i=0}^\infty ix^i is x / (1-x)^2 .
So the sum of interest is 0.001 times (0.999) / (0.000001), which is exactly 999.