Question

Suppose ancient Romans had a 0.2 chance of dying in each of the following age intervals: [0, 2], [2, 10], [10, 30], [30, 70] and [70, 90].
To keep calculations straightforward, let's assume that, conditional on dying in any one of those five buckets, age at death was uniformly distributed across the interval. That means that, conditional on dying between age 0 and 2, the average Roman lived to be 1 year old; conditional on dying between age 2 and 10, the average was 6 years; et cetera. Under these (fictitious) numbers, life expectancy at birth was only 31.4 years, due largely to child mortality.

What was the life expectancy (i.e. expected age at death) of an ancient Roman who was still alive at age 30? It's much more than 31.4 years — can you figure out why?

Answer 1

65 As for the reason the average life expectancy of a Roman who reaches the age of 30 being so much higher than the average expectancy overall, that's simply a matter of taking the average of 50 and 80, verses the average of 1,6,20,50,80. Let's illustrate that by calculating the average life expectancy of a Roman at birth, and after age 30. For birth, there's 5 ranges, each of which has the same probability. They are [0,2]: Midpoint = 1. Probability = 0.2. Product = 1*0.2 = 0.2 [2,10]: Midpoint = 6. Probability = 0.2. Product = 6*0.2 = 1.2 [10,30]: Midpoint = 20. Probability = 0.2. Product = 20*0.2 = 4 [30,70]: Midpoint = 50. Probability = 0.2. Product = 50*0.2 = 10 [70,90]: Midpoint = 80. Probability = 0.2. Product = 80*0.2 = 16 Sum = 0.2 + 1.2 + 4 + 10 + 16 = 31.4 But upon reaching 30, there is no longer a mere 0.2 probability for those last 2 slots. The chart looks like [30,70]: Midpoint = 50. Probability = 0.5. Product = 50*0.5 = 25 [70,90]: Midpoint = 80. Probability = 0.5. Product = 80*0.5 = 40 Sum = 65 If you look at each possible range of ages, the actual life expectancy is at birth: 31.4 years after age 2: 39 years after age 10: 50 years after age 30: 65 years after age 70: 80 years