A man purchases a life insurance policy on his $40$th birthday. The policy will pay $5000$ only if he dies before his $50$th birthday and will pay $0$ otherwise. The length of lifetime, in years, of a male born the same year as the insured has the cumulative distribution function: $$F(t) \ \ = \ \ 1 - e^{\frac{1-1.1^t}{1000}} \ \ \ \ \ \ \ \ \ \ t > 0$$ The question is asking for the expected payment, and the answer key says $347.96$. How come the answer is not the following? $$EX \ \ = \ \ \int_{40}^{\, 50} 5000 \cdot f(x) \, dx \ \ = \ \ 5000 [ \, F(50) - F(40) \, ] \ \ \approx \ \ 332.89$$

Here is the original question:

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    $\begingroup$ Presumably you have to take into account that he has not died in the first $40$ years by dividing by a suitable number $\endgroup$ – Henry Apr 23 '17 at 23:26

You are considering the unconditional probability that the person lives between 40 and 50. You need to take into account that he has lived 40 years already. We have \begin{align*} 5000P(X\leq 50|X\geq 40)&= 5000\cdot\frac{P(40\leq X\leq 50)}{P(X\geq 40)}\\ &=5000\cdot \frac{F(50)-F(40)}{1-F(40)}\\ &\approx 347.96 \end{align*}

  • $\begingroup$ Thanks for the response. Can I ask if you were to able to tell immediately that the question is asking for the conditional expectation? I am practicing this question again and fell for the same mistake. I appended a copy of the original question to my post. $\endgroup$ – Andy Tam Jul 9 '17 at 15:30
  • $\begingroup$ Yes, immediately. The first sentence "A man purchases a life insurance policy on his 40th birthday" implies "Given that the man purchases a life insurance policy on his 40th birthday, calculate the expected payment under this policy". Your notation is a little sloppy, but I can see why you thought it was your answer since the bold part is not written. You just have to try to remember or determine that it's implied. By the way, you don't need to post images. Just try to type the question as accurately as possible. $\endgroup$ – Em. Jul 10 '17 at 5:04

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