Hi, I follow this question and solution. I just had a question regarding the E(X) and E(X^2) calculated values. So because the E(J) (expected value) for say the junior debt holder is a function of another random variable L, where L~UNI(0,50), which is the loss function (L~UNI(0,100) rescaled for just the junior debt holder, when finding the expected value it's E(J) = 0.75*54+0.25*0.5*0 + 0.25*0.5*(50-E(L)). This will give the same answer as IFoA as they model L as a proportion whereas I model as a total value. So my question is this - if we are finding the expected value of a r.v. that is a function of another r.v., we just when calculating the expected value of original r.v. we use the law of the unconscious statistician (LOTUS) ie E[g(X)] = Sum,g(xk)*Px(xk), where Y=g(X)? Lastly if we didn't have a uniform distribution for the loss function ie L~UNI(0,100) ie a complex unsymmetrical distribution, how would we rescale? Would we just integrate? Thanks, Darragh
Hi Yes, the expectation of a function of a random variable follows this pattern. Very commonly seen in Chapter 2, where the expected utility is given by: E[U(W)] = sum(P(W=w_i) * U(w_i)) The distribution of W is known, whereas the distribution of U(W) is unknown (and not required).
Yeah thanks never thought of it like that but its the same idea thing I'm describing. Thanks for your help on this. Darragh
