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Three quesitons needs help

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  • Three quesitons needs help

    Dear all, please help me to solve below questions. Thank you.:confused-:

    Last edited by amywliang; May 14 2009, 06:26 AM.

  • #2
    (1) the model for the amount of damage is a uniform distribution between 200 to 1200. The policy of insurance pays the amount of damage up to 400. The remaining of amount is covered by supplement policy, which pays up to 500. Let Y denote the supplement policy. Find V(y)

    So let X be the loss (the amount of damage) which is a uniform dist (200,1200) and let Y be the supplemental policy. Thus the main policy will pay a maximum of 400; meaning that it will pay X if X<400 and will pay 400 if X>400.

    The remaining amount is covered by the supplemental policy, which will pay up to 500 EXTRA loss. This means that it will pay anything in excess of 400 up to 400+500 = 900. Thus all you need to do is find the expected value of Y, which is done by integrating the loss at a deductible of 400 (X-400) multiplied by the joint pdf (1/1000) from 400 to 900; then adding that to the integral of the loss (X) multiplied by the joint pdf (1/1000) from 900 to the highest limit the loss can go, which is 1200. You should get the answer.


    (2) F(x)= (Y*X^1/2)/ 1000 0<x<100 and 0<y<100
    find p( 10<x<75, 10<y<75)

    So I believe that the actual CDF is F(x,y)= (Y*X^1/2)/ 1000 0<x<100 and 0<y<100 since there is a y-variable in the equation and thus must have been a joint CDF. To answer this question is easy. Simply input the following in the CDF: F(75, 75) - F(10, 10) and you should get the answer.

    To do this, you would take the double integral of F(x,y) from 0 to 75 for both x and y, then subtract the double integral of F(x, y) from 0 to 10 for both x and y. I hope this is clear.


    (3)smith is already 70 years old, the density function of remaining of lifetime is as below, find the probability of he will die after 10 years. given smith will be alive 5 years
    F(t)= (1-1.0^t)/1.1 t>0

    Basically you need to know how to interpret and understand this question. Read the question carefully. Digest what they have told you, digest what they are asking you to do. Read it a few times if you have to.

    They give you the density function of his remaining lifetime. So we don't really care about the fact that he is already 70 years old - it doesn't matter. We are only concerned about his remaining lifetime. He could be 10, 20, 35, 55 for all we care. The fact that he is already 70 doesn't make a difference.

    Now, we are given that he will be alive after 5 years. What is the probability he will die after 10 years?

    Thinking about it, we should realize that they are actually asking, what is the probability that Smith will NOT DIE between the ages of 70+5=75 and 70+10=80. This would make use of the density function that they provide, which is essentially the probability of the years that he will NOT DIE (i.e., remain alive, remaining lifetime).

    With that analysis, before even thinking of applying any brute force method, it should be clear that all you need to do is take the pdf of the cdf equation they give and integrate from 5 to 10, then divide it by the integral of 5 to infinity (conditional probability concept - ask me if you are not clear on this step).

    However, in order to avoid messy integration, you could just notice that integrating the pdf of the equation from 5 to 10 is the same as inputting F(10) - F(5) into the equation, then dividing it by the survival function, which is 1 - F(5).

    That should also give you the correct answer.

    I hope this helps


    • #3
      Thank you

      :smiloe: [B]Thank you for your helping. It's so nice to have someone to help me.[/B]
      Last edited by amywliang; May 14 2009, 06:26 AM.