Actuarial Outpost Darth Vader Rule Question
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#1
05-05-2018, 02:24 AM
 fjmvasa CAS SOA Join Date: Apr 2018 Posts: 2

Hi everyone! I would like to ask if the Darth Vader also applies for x < 0? Since there is a rule for a < x < infinity.
#2
05-05-2018, 03:35 AM
 Michael Mastroianni SOA Join Date: Jan 2018 Posts: 8

If $X$ is a random variable defined on $[a,\infty)$ and $a<0$ then $X-a$ is defined on $[0,\infty)$ right? So $X-a$ is nonnegative and you can use the rule:

$\displaystyle E[X-a]=\int_{0}^{\infty} [1-F_{X-a}(t)] \, dt=\int_{0}^{\infty} [1-P(X-a\leq t)] \, dt=\int_{0}^{\infty} [1-P(X\leq t+a)] \, dt=\int_{a}^{\infty} [1-P(X\leq x)] \, dx$

Where the last part uses substitution: $x=t+a$

But also from linearity we know $\displaystyle \int_{a}^{\infty} [1-F_{X}(x)] \, dx=E[X-a]=E[X]-a$

Add $a$ to both sides for: $\displaystyle E[X]=a+\int_{a}^{\infty} [1-F_{X}(x)] \, dx$

So in this limited case it does extend to negative values. I hope that helps.
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Michael Mastroianni, ASA
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#3
05-06-2018, 10:03 PM
 fjmvasa CAS SOA Join Date: Apr 2018 Posts: 2

Thank you very much Michael!