Question

A financial analyst has found out that policyholders are four times as likely to file two claims as to file four claims. If the number of claims filed has a poison distribution, what is the variance of the number of claims filed

Answer 1

1. We want to find $\lambda$ for the Poisson RV from $P(X=2)=4P(X=4)$

$$P(X=k)={e^{-\lambda}\lambda^k\over k!}$$

$$P(X=2)={e^{-\lambda}\lambda^2\over 2!}=4P(X=4)=4\cdot{e^{-\lambda}\lambda^4\over 4!}$$

$$\lambda^2={4!\over4\cdot2!}=3$$

$V(X)=\sigma^2=\lambda=\sqrt{3}$