Q. 6.43 - A First Course in Probability

Question

An insurance company supposes that each person has an accident parameter and that the yearly number of accidents of someone whose accident parameter is λ is Poisson distributed with mean λ. They also suppose that the parameter value of a newly insured person can be assumed to be the value of a gamma random variable with parameters s and α. If a newly insured person has n accidents in her first year, find the conditional density of her accident parameter. Also, determine the expected number of accidents that she will have in the following year.

Step-by-Step Solution

Answer

Conditional density:

$f_{\lambda \mid N}(l\mid n)=\frac{a^s}{n!·p_n·\Gamma \left(s\right)}{l}^{n+s-1}{e}^{-l(\alpha +1)}$

Expected number of accidents:

$E(\lambda \mid N=n)=\frac{n+s}{\alpha +1}$

Step 1: Given information

The accident parameter is $\lambda$ is Poisson distributed with mean $\lambda$.

$\lambda$ is a random variable with distributed Gamma $(s,\alpha )$.

The newly insured person has n accidents in her first year.

Step 2: Explanation

Let N be the random variable that marks the number of accidents of some person in a certain year.

According to the statement, the density function of $\lambda$,

$f_{\lambda }\left(l\right)=\frac{{\alpha }^s}{\Gamma \left(s\right)}{l}^{s-1}{e}^{-\alpha l}$

Now, with N = n

We need o find the conditional density of $\lambda$.

Then using the Bayesian formula,

$f_{\lambda \mid N}(l\mid n)=\frac{P(N=n\mid \lambda =l)f_{\lambda }\left(l\right)}{P(N=n)}$

Now define

$P(N=n)=p_n$

we have

$\frac{P(N=n\mid \lambda =l)f_{\lambda }\left(l\right)}{P(N=n)}=\frac{1}{p_n}·\frac{l^n}{n!}{e}^{-l}·\frac{{\alpha }^s}{\Gamma \left(s\right)}{l}^{s-1}{e}^{-\alpha l}$

such that

$f_{\lambda \mid N}(l\mid n)=\frac{a^s}{n!·p_n·\Gamma \left(s\right)}{l}^{n+s-1}{e}^{-l(\alpha +1)}$

with parameters $n+s and \alpha +1$,

$\lambda /N=n$ has Gamma distribution.

Moreover, the expected number of accidents that she will have in the following year,

$E(\lambda \mid N=n)=\frac{n+s}{\alpha +1}$

Where the formula for the expected value of gamma distribution has been used.