Q - "The number of claims, X, arising on each policy in a certain portfolio depends on another random variable Y. X is considered to follow a poisson distribution wuth mean Y. The variable itself is assumed to have a gamma distribution with parameters (a,b). Find expressions for E(X) and E(X^2) using conditional moments" In the revision notes, the first line (bold part) has been interepreted as X|Y ~ Poi (Y) Why? The obvious interpretation seems to be X ~ Poi (Y).
Since Y is a random variable we can't say X~Poi(Y) as we don't know what value Y is taking. Hence, we define it as X|Y=y ~ Poi(y) as the Y value is fixed as the value y and so the distribution is now well defined. From the point of calculating the answers it makes no difference - but it's just the correct way of doing it.
