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Math Help - Density Function

  1. #1
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    Density Function

    With Mr. Fantastic's help, I realized how to do my other set of problems. This time, I only have one problem. I will put the problem in quotes. Then my idea of how to do the problem after.

    The total individual claim amount has a density function f(x)= (1/1000)e^-(x/1000); x(greater than or equal to)0. The premium is set at 100 over the expected claim amount. A portfolio consists of 100 independent policies with this claim distribution. Calculate the probability that the total claim amount exceeds total premiums collected.
    Should I first treat it as a deductible problem and find the mean of that. By this I mean:
    The integral (min=0 max =100) of (1/1000)e^-(x/1000) + Integral (min=100 max = infinity) of (x-100)(1/1000)e^-(x/1000). This will give me the expected value of the premiums for each claim. I would multiply this by 100 to get the total expected premiums. I dont think I can treat this as a normal distribution because the problem does not say that it is normal.

    Thanks in advance for the help.
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  2. #2
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    Quote Originally Posted by PensFan10 View Post
    With Mr. Fantastic's help, I realized how to do my other set of problems. This time, I only have one problem. I will put the problem in quotes. Then my idea of how to do the problem after.



    Should I first treat it as a deductible problem and find the mean of that. By this I mean:
    The integral (min=0 max =100) of (1/1000)e^-(x/1000) + Integral (min=100 max = infinity) of (x-100)(1/1000)e^-(x/1000). This will give me the expected value of the premiums for each claim. I would multiply this by 100 to get the total expected premiums. I dont think I can treat this as a normal distribution because the problem does not say that it is normal.

    Thanks in advance for the help.
    Cheating and having a look at Exponential distribution - Wikipedia, the free encyclopedia instead of doing the integration, E(X) = 1000.

    Therefore premium = E(X) + 100 = 1100.

    Let U = X_1 + X_2 + \, .... \, + X_{100}.

    The pdf for U is given here Exponential distribution - Wikipedia, the free encyclopedia (towards the end): U ~ \Gamma (100, 1000) (the definition is here: Gamma distribution - Wikipedia, the free encyclopedia).

    Use the pdf to calculate \Pr(U > 1100).
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  3. #3
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    Uhh. Once again, you are a big help. Thank you.
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