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 X, if Pr(X≥a)>0, E[X∣X≥a]≥E[X].
Let c=Pr(X≤a) E[X]=∫−∞∞xPr(X=x)dx=∫−∞axPr(X=x)dx+∫a∞xPr(X=x)dx=∫−∞∞xPr(X=x∣X≤a)Pr(X≤a)dx+∫−∞∞xPr(X=x∣X≥a)Pr(X≥a)dx=c∫−∞∞xPr(X=x∣X≤a)dx+(1−c)∫−∞∞xPr(X=x∣X≥a)dx=cE[X∣X≤a]+(1−c)E[X∣X≥a]
If a=λb+(1−λ)c, where λ∈[0,1], then a≤max(b,c). Because E[X∣X≤a]≤a≤E[X∣X≥a], E[X]≤E[X∣X≥a].
In fact, one can easily modify the above proof and prove the next theorem:
For any real random variable X, if x≥y and Pr(X≥x)>0, then E[X∣X≥x]≥E[X∣X≥y].
A heuristics conclusion: The longer you lived, you expect to live longer.