You can use either 0 to R or 0 to F(R). What matters is you carefully note the terms of the limit. In the first case you're saying x goes from 0 to R, whereas in the second case you're saying F(x) goes from 0 to F(R). It's the classic Calc II switching the limits of integration.
The Fisher text presents both the size and layer methods with limits in terms of the size of loss, x. This means the size method is less intuitive to read because the integral and the presentation of the limits don't match mathematically :(
In that case, then the BattleWiki doesn't seem to be consistent. I have the same question as OP. The examiner's report has the limits from 0 to R for both versions of integral dF(x) or f(x)dx. So should the BattleWiki example be integrating from D to L instead?
The examiner's report is mathematically inconsistent in sample 2 because it plays loose with the notation in the limits. The same is true of the Fisher text on page 52.
Remember, dF(x) means a small change in F(x) and F(x) is only defined between 0 and 1. D and L are loss dollars so highly unlikely to be in that range. The correct bounds are F(D) to F(L).
dF(x) is equivalent to f(x)dx where now the small change is in x, the loss size. So in this formulation it's correct to integrate between D and L.
Comments
You can use either 0 to R or 0 to F(R). What matters is you carefully note the terms of the limit. In the first case you're saying x goes from 0 to R, whereas in the second case you're saying F(x) goes from 0 to F(R). It's the classic Calc II switching the limits of integration.
The Fisher text presents both the size and layer methods with limits in terms of the size of loss, x. This means the size method is less intuitive to read because the integral and the presentation of the limits don't match mathematically :(
In that case, then the BattleWiki doesn't seem to be consistent. I have the same question as OP. The examiner's report has the limits from 0 to R for both versions of integral dF(x) or f(x)dx. So should the BattleWiki example be integrating from D to L instead?
The examiner's report is mathematically inconsistent in sample 2 because it plays loose with the notation in the limits. The same is true of the Fisher text on page 52.
Remember, dF(x) means a small change in F(x) and F(x) is only defined between 0 and 1. D and L are loss dollars so highly unlikely to be in that range. The correct bounds are F(D) to F(L).
dF(x) is equivalent to f(x)dx where now the small change is in x, the loss size. So in this formulation it's correct to integrate between D and L.