The number of claims made by an insured individual in a year is a random variable following a Poisson distribution with unknown mean .

The number of claims made by an insured individual in a year is a random variable following a Poisson distribution with unknown mean θ. Based on previous experience, an insurer's prior feeling about θ is that it can be represented by an exponential random variable with mean 1. Determine two bayesian estimates of θ given that the mean of the annual number of claims of this individual over the last 10 years is x_10 = 3.

You may use the fact that, if α ≥ 1 then the mode of the Gamma(α, λ) distri- bution is given by (α−1)/λ.