Q8 (i) and (ii) 2015 Oct pastpaper

[Chang Liu]

For Q8 (i), I don't really understand the prove, it seems any function Y= F(x) with 0<Y<1 can be proved to be U(0,1) random variable ?? Can any explain this in details as the exam report is very short.

For Q8 (ii). I understand the solution is trying to find value of a and b to standardize the Ln(S2).
BUT my question is how does this prove the cumulative distribution function of std normal distribution is a U(0,1) random variable?
I don't see the link between the solution and the question .. Could anyone explain it ?

Many thanks!

 

[Mark Mitchell]

You should have seen the result in Q8(i) before in something like CT3 or CT6. It's a general result for the distribution function of a continuous RV. The short proof relies specifically on the definition of a distribution function for a random variable, X: F(x) = P(X<x). Other than that, there's little to add to what's given in the examiners report.

Part (ii) of the question relies on part (i).

To see how it works, let X = (lnS2 - a)/b. This is a continuous RV (as lnS2 is a normal RV, and hence continuous). We know from (i) that Y = F(X) ~ U(0,1) if F is the distribution function of X. We're considering Phi(X), where Phi is the distribution function of the standard normal RV, so to use the result in (i), we need X to be a standard normal RV, so we choose a and b accordingly.

 

[Chang Liu]

You should have seen the result in Q8(i) before in something like CT3 or CT6. It's a general result for the distribution function of a continuous RV. The short proof relies specifically on the definition of a distribution function for a random variable, X: F(x) = P(X<x). Other than that, there's little to add to what's given in the examiners report.

Part (ii) of the question relies on part (i).

To see how it works, let X = (lnS2 - a)/b. This is a continuous RV (as lnS2 is a normal RV, and hence continuous). We know from (i) that Y = F(X) ~ U(0,1) if F is the distribution function of X. We're considering Phi(X), where Phi is the distribution function of the standard normal RV, so to use the result in (i), we need X to be a standard normal RV, so we choose a and b accordingly.

Thanks a lot Mark!