Question

Let the number \(X\) of accidents have a Poisson distribution with mean \(\lambda \theta\). Suppose \(\lambda\), the liability to have an accident, has, given \(\theta\), a gamma pdf with parameters \(\alpha=h\) and \(\beta=h^{-1} ;\) and \(\theta\), an accident proneness factor, has a generalized Pareto pdf with parameters \(\alpha, \lambda=h\), and \(k .\) Show that the unconditional pdf of \(X\) is $$ \frac{\Gamma(\alpha+k) \Gamma(\alpha+h) \Gamma(\alpha+h+k) \Gamma(h+k) \Gamma(k+x)}{\Gamma(\alpha) \Gamma(\alpha+k+h) \Gamma(h) \Gamma(k) \Gamma(\alpha+h+k+x) x !}, \quad x=0,1,2, \ldots $$ sometimes called the generalized Waring pmf.

Short answer

The unconditional pdf of \(X\) can be obtained by writing down the joint distribution of \(\lambda\) and \(\theta\) first, computing the marginal distribution of \(X\) by integrating out \(\lambda\) and \(\theta\), and simplifying the resultant expression using algebraic simplifications and properties of gamma function. The final expression we get should match the given expression in the problem statement, which is known as the generalized Waring pmf.

Step-by-step solution

Understand the Problem

This problem is about understanding the relationships between different probability distributions. We have a Poisson, gamma, and generalized Pareto distribution, and we need to find the unconditional pdf of the Poisson distribution.

Formulate the Joint Distribution of \(\lambda\) and \(\theta\)

We have to write down the joint distribution of \(\lambda\) and \(\theta\), which is the product of their respective densities. That is:\[ p(\lambda, \theta | X) = p(\lambda | \theta, X) p(\theta | X) \]Using the given gamma and Pareto pdf expressions for the respective distributions, substitute them into this formula.

Write Down the Marginal Distribution of \(X\)

Next, since we want to find the unconditional pdf of \(X\), we need to write down the marginal distribution of \(X\) by integrating out \(\lambda\) and \(\theta\) from the joint distribution. We do this by setting up and calculating the following integral over the support of \(\lambda\) and \(\theta\):\[ p(X) = \int \int p(\lambda, \theta | X) d\lambda d\theta \]

Simplify the Resulting Expression

The expression obtained in the previous step will likely be quite complex. Use algebraic simplifications, properties of gamma functions, and substitutions wherever possible to simplify the integral. Eventually, we want to show that this simplifies to the given expression for the unconditional pdf of \(X\).