Question

An actuary has discovered that policyholders are three times as likely to file two claims as to file four claims .. If the number of claims filed has a Poisson distribution, what is the variance of the number of claims filed?

Answer 1

Let X denote the number of claims. If X has a Poisson distribution, then, based on the given information you can write:

`P(X=2)=3P(X=4)`

Then, by using the Poisson distribution:

`P(X=x)=(e^(-λ)λ^x)/(x!)`

you can write:

`(e^(-\lambda)\lambda^2)/(2!)=3((e^(-\lambda)\lambda^4)/(4!))`

Now, it is neccesary to determine the solution for λ from the previous equation. Cancel e^-λ both sides, divide by λ^2 both sides and the apply square root and solve for λ:

`\begin{gathered} (λ^2)/(2)=3(λ^4)/(24) \\ λ^2=4 \\ λ=\sqrt[\placeholder{⬚}]{4}=2 \end{gathered}`

Now, consider that the variance in a Poisson distribution is given by

σ^2 = λ

Hence, the variance of the number of claims filed is 2