# Math Help - Density Function

1. ## 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.

2. Originally Posted by PensFan10
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 $\Pr(U > 1100)$.

3. Uhh. Once again, you are a big help. Thank you.