Insurance Savings Practice Problem
I'm really struggling with the exponential Insurance Savings example. I think it's not clear to me why we need the pdf of the exponential distribution. Why can't we just take the derivative of the CDF like we did in the Insurance Charge example? I can see it doesn't seem to work out the same way if I try to do that, but just confused on why.
Comments
In both the insurance charge example and the insurance savings example we are looking at the entry ratios, A/E, where A is the actual loss and E is the expected loss. For ease of notation we let Y = A/E.
The formulas for calculating the respective charge and savings involve the derivative of the cumulative distribution function of Y. Let F be the c.d.f. for Y, the formulas involve integrating over dF(y), which is the p.d.f. of Y.
In both the charge and savings example, if you know the c.d.f. for Y you can differentiate it and apply the formulas. The savings example is more complicated because, at least to me, at a first glance it isn't clear what the c.d.f. for Y really it. This is why we use the side discussion about calculating the p.d.f. of Y = aX+b when we know the p.d.f. of X. Perhaps it would have been better to write Z = aX+b so Y is only used once as defined in the question.
To answer your question: Yes, we can just take the derivative of the CDF like we did in the insurance charge example. However, that means you must be able to know what the correct CDF is.
Once you've correct got this, you apply it in the integral version of the insurance savings formula.
P.S.
We decided to give more detail than Fisher does in their text so you know how they arrived at their answer.