Hello
The overarching structure of the models is the same in the sense that we have:
1. A Poisson distribution representing the number of infections for each individual patient (ie a collective risk model for each patient based on their Poisson parameter)
2. 100 patients in each study
The different between the two scenarios comes down to the Poisson parameter. In parts (i) and (ii), the Poisson parameter for each patient is either 0.1, 0.3 or 0.9 with the probabilities given in the table. These parameters can differ across patients, so we might have:
0.1, 0.3, 0.3, 0.1, 0.9, 0.1, 0.1, 0.3, .... etc for the 100 patients.
Or instead we might have:
0.3, 0.9, 0.9, 0.3, 0.3, 0.1, 0.9, ... etc for the 100 patients
The possible scenarios here are all the combinations of 0.1, 0.3 and 0.9 across the patients. When calculating the aggregate claims for all 100 patients, we take into account the variation in the Poisson parameter across patients based on the given distribution.
In part (iii), the Poisson parameter is the same for each patient, we just don't know which of the two it is. So we have two possible scenarios here, either all patients have parameter 0.2 or all patients have parameter 0.4. These are two 'extremes' compared to parts (i) and (ii) before, where we consider all the possible mixes of parameters across the patients. This contributes to why we end up with a larger variance for the aggregate claims in part (iii), even though the mean is the same.
Variance calculations
For part (ii)(c), we have from page 16 of the Tables:
Var(Si) = var(E[Si | lambda_i]) + E[var(Si | lambda_i)]
= var(lambda_i * m1) + E[lambda_i * m2]
= m1^2 * var(lambda_i) + m2 * E[lambda_i]
= 250^2 * 0.08 + (200 + 250^2) * 0.3 = 23810
The overall variance for the aggregate claims is then 100 * 23810 = 2,381,000
For part (iii), we have:
Var(S) = E[var(S | lambda)] + var(E[S | lambda])
= E[n * lambda * m2] + var(n * lambda * m1)
= 100 * 0.3 * (200 + 250^2) + 100^2 * 250^2 * 0.01 = 8,131,000
It looks like there is an error in the Examiners' report here, for m2 I think it uses 200 instead of 200 + 250^2.
The reason we can't take the same approach in part (iii) as we did in part (ii)(c), ie calculating the individual variances of Si and multiplying by 100, is that the Si's here are not independent, since they are all driven by the same unknown value of lambda. If lambda is known then they are independent (so independent conditional on lambda) and hence we can perform the calculations above.
Hope this helps!
Andy
Click to expand...
