To calculate the probability density function (pdf) of the posterior distribution, we need to use Bayes' theorem. Bayes' theorem allows us to update our prior understanding of the distribution based on new data.
Given that the claim sizes follow a Poisson distribution with parameter λ = 0, we can use the prior understanding of the distribution, which has the pdf:
h(λ) = 1 for 1 < λ < 0.02
Now, we observed two claims, one of size 1 and another of size 2. We want to calculate the pdf of the posterior distribution, which represents our updated understanding of the distribution after observing these claims.
To do this, we need to find the likelihood function, which represents the probability of observing the data given a specific value of λ. In this case, the likelihood function is the product of the probabilities of observing claim sizes 1 and 2 under the Poisson distribution with parameter λ = 0. So, the likelihood function is:
L(λ) = P(X = 1; λ) * P(X = 2; λ)
Since claim sizes follow a Poisson distribution, the probabilities can be calculated using the Poisson probability mass function (pmf):
P(X = k; λ) = (e^(-λ) * λ^k) / k!
For k = 1 and k = 2, we have:
P(X = 1; λ) = (e^(-0) * 0^1) / 1! = 0
P(X = 2; λ) = (e^(-0) * 0^2) / 2! = 0
So, the likelihood function becomes:
L(λ) = 0 * 0 = 0
Now, we can apply Bayes' theorem to calculate the posterior distribution. Bayes' theorem states:
P(λ | X) = (L(λ) * h(λ)) / ∫[0.02,1] (L(λ) * h(λ)) dλ
Since the likelihood function is 0 and the prior pdf h(λ) is constant for 1 < λ < 0.02, the numerator of Bayes' theorem becomes 0. Therefore, the posterior distribution is also 0 for 1 < λ < 0.02.
In conclusion, the probability density function of the posterior distribution is 0 for 1 < λ < 0.02.