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Math Help - Conceptual difficulties with normal distribution

  1. #1
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    Conceptual difficulties with normal distribution

    So I have to solve this problem. I cannot pose the question I have without illustrating the entire problem so here it is:

    N = annual medical expenses distributed normally with mean $3500 and deviation $1200.

    Jane has the option of prepaying ANY amount 'q' such that:
    1] her company will pay the first 'q' dollars of medical expenses but anything that exceeds she will have to bear. But if N is less than q, she doesnt get back her money.
    2] Her taxable income is (Salary - q). So basically she gets a tax benefit for paying 'q'. Tax rate is 38%.

    (a) Determine q* that maximizes expectation of net income.
    Her net income = Taxable income - Tax - excess medical expenses.
    =(Salary - q*) - 0.38(Salary-q*) - MAX{0,(N - q*)}
    = 0.68xSalary - 0.68q* - MAX{0,(N-q*)}
    Since Salary is a constant, it doesnt really matter (I'm guessing). I use solver in Excel to maximize and it gives me q* = 3500, which is the mean of N anyway. Does this make intuitive sense?

    What I'm confused about is Expectation(N - q*). Here, N is a normally distributed random variable but q* is a constant. Using the rule,
    E(N-q*) = E(N) - E(q*) = 0 (coz both of them are 3500).

    I was happy with my answer till I saw part (b)

    (b) What is the expectation of the amount of medical expenses she will have to pay next year?
    Intuitively, this is E(N-q*), but that is ZERO! That doesn't quite make sense because there is a probability that N>q.

    Please please please help!
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  2. #2
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    Quote Originally Posted by indian View Post
    So I have to solve this problem. I cannot pose the question I have without illustrating the entire problem so here it is:

    N = annual medical expenses distributed normally with mean $3500 and deviation $1200.

    Jane has the option of prepaying ANY amount 'q' such that:
    1] her company will pay the first 'q' dollars of medical expenses but anything that exceeds she will have to bear. But if N is less than q, she doesnt get back her money.
    2] Her taxable income is (Salary - q). So basically she gets a tax benefit for paying 'q'. Tax rate is 38%.

    (a) Determine q* that maximizes expectation of net income.
    Her net income = Taxable income - Tax - excess medical expenses.
    =(Salary - q*) - 0.38(Salary-q*) - MAX{0,(N - q*)}
    = 0.68xSalary - 0.68q* - MAX{0,(N-q*)}
    Since Salary is a constant, it doesnt really matter (I'm guessing). I use solver in Excel to maximize and it gives me q* = 3500, which is the mean of N anyway. Does this make intuitive sense?

    What I'm confused about is Expectation(N - q*). Here, N is a normally distributed random variable but q* is a constant. Using the rule,
    E(N-q*) = E(N) - E(q*) = 0 (coz both of them are 3500).

    I was happy with my answer till I saw part (b)

    (b) What is the expectation of the amount of medical expenses she will have to pay next year?
    Intuitively, this is E(N-q*), but that is ZERO! That doesn't quite make sense because there is a probability that N>q.

    Please please please help!
    E(aX + b) = a E(X) + b. So:

    E(N - q*) = E(N) - q* = 0.

    This doesn't mean she pays nothing .... In fact, Pr(N - q* > 0) = 1.2 ......
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