April 2010 Q9

[Darragh Kelly]

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

 

[Steve Hales]

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).

 

[Darragh Kelly]

Yeah thanks never thought of it like that but its the same idea thing I'm describing.
Thanks for your help on this.
Darragh