# Thread: Conceptual difficulties with normal distribution

1. ## 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.

2. Originally Posted by indian
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.