[Actuarial] Poisson Parameter with Gamma Prior

In Bayesian credibility, if the frequency of claims follows a Poisson distribution with parameter/mean (lambda), and lambda has a Gamma prior distribution, there are some shortcuts to determine the posterior distribution of lambda as well as its mean and variance based on observations. 

The formula for the posterior distribution has a numerator,  

    Pr( N | lambda = lambda_i ) * Pr( lambda = lambda_i ),

where N is the number of observed claims. The first term is a Poisson probability function and the second is a Gamma density function. The product is in the form of 

    c * lambda^a * exp(-b * lambda),

where a, b and c are constants. The integral of it over the domain of lambda makes the denominator, by which we divide the numerator to get the posterior distribution. The integral yields another constant, and there is a shortcut to get its value without actually working out the integration, which is simply calculating 

    c * (1/b)^(a+1) * a!. 

Now we have both the numerator and the denominator, the posterior distribution of lambda can therefore be determined. 

To get the mean and variance of the posterior distribution, we need the parameters alpha and gamma (or 1/theta) from the Gamma prior. 

Since the posterior is also a Gamma distribution, it has two parameters, denoted by alpha_new and gamma_new. To transform alpha to alpha_new, we calculate

    alpha_new = alpha + sum(x_i),

where x_i denotes the observed numbers of claims, similarly, 

    gamma_new = gamma + n, 

where n is the number of risks. 

The posterior mean is therefore 

    alpha_new/gamma_new,

and the variance is 

    alpha_new/(gamma_new^2).

Exponential distribution is a special case of Gamma distribution with alpha equal to 1, so in problems that the prior has an exponential distribution, we can solve them with the same methods discussed above.