### Poisson model with a constant parameter

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A brilliant paediatrician has such excellent diagnostic skills that without resorting to expensive and sometimes painful tests, she rarely misdiagnoses what ails her patients. Her overall average misdiagnosis rate of 1 per 1,000 consultations is all the more remarkable given that many of her patients are often too young to describe their symptoms adequately-when they can describe them at all; the pae- diatrician must therefore often depend on indirect information extracted from the parents and guardians during clinic visits. Because of her other responsibilities in the clinic, she must limit her patient load to precisely 10 per day for 325 days a year. While the total number of her misdiagnoses is clearly a Poisson random variable, the Poisson parameter λ = ηt is not constant because of the variability in her patient population, both in age and in the ability of parents and guardians to communicate effectively on behalf of their non-verbal charges. If λ has a gamma distribution with α = 13,β = 0.25,

(i) Determine the probability that the paediatrician records exactly 3 misdiagnoses in a year; determine also the probability of recording 3 or fewer misdiagnoses.

(ii) Compare these probabilities with the corresponding ones computed using a stan- dard Poisson model with a constant parameter.

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