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Sept 2008 Q5 - Conditional Moments

J

Jammy

Member
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.
 
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