Submitted by pyhali on Thu, 01/11/2018 - 18:18
Forums:
In the first part of question 2 of the STEP 3 stats questions (https://maths.org/step/sites/maths.org.step/files/s2s3/Stats3_hints.pdf) it gives general PGFs G(t) and H(t) for two random variables and gives another random variable, Y, defined in terms of the previous two. It then says "Given that the pgf of Y is G(H(t)), show that...". The last part of the question gives three specific random variables which have the same properties of the generic random variables at the start except that the pgf of Y being G(H(t)) is not clear. can you assume that the pgf of Y is G(H(t)) in this case or should it be proved?
Q2
You need to explain that the conditions in the first part of the question are satisfied in the last situation (so that $X_i$ are independently and identically distributed and are independent of $N$ etc). I would start by clearly explaining what $X_i$, $N$ and $Y$ are. As long as the conditions are met, you can then assume the given result for the pgf of $Y$.