While discussing conditional probability, someone said the following:
The expected life expectancy of some country is 70, and there exist people who die at every age before 70. What is the expected life expectancy for a 60 year old?
Most people would answer 10. However, he continuous:
It could be 10, but for many distributions, it’s likely more than that. You can convince yourself by thinking about the expected life expectancy for a 80 year old.
The quote above would follow directly from the proof of the following theorem:
For any real random variable , if , .
If , where , then . Because , .
In fact, one can easily modify the above proof and prove the next theorem:
For any real random variable , if and , then .
A heuristics conclusion: The longer you lived, you expect to live longer.