# How long do you expect to live?

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=∫_{−∞}xPr(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.