Let X denote the number of claims. If X has a Poisson distribution, then, based on the given information you can write:
[tex]P(X=2)=3P(X=4)[/tex]Then, by using the Poisson distribution:
[tex]P(X=x)=\frac{e^{-λ}λ^x}{x!}[/tex]you can write:
[tex]\frac{e^{-\lambda}\lambda^2}{2!}=3(\frac{e^{-\lambda}\lambda^4}{4!})[/tex]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:
[tex]\begin{gathered} \frac{λ^2}{2}=3\frac{λ^4}{24} \\ λ^2=4 \\ λ=\sqrt[\placeholder{⬚}]{4}=2 \end{gathered}[/tex]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
