Please help out

Question

A financial analyst has found out that,policy holders are four 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 Expected Value of the numbers of claims filed.

Answer 1

Let X = number of claims filed by a particular policy holder

We are given than P(X = 2) = 4*P(X = 4). Let's plug in the Poisson pmf formula [P(X = x) = λ^x*e^-λ/x!] and solve for λ:

λ²*e^-λ/2! = 4*λ⁴*e^-λ/4!

the e^-λ cancels out on both sides. If we divide both sides by λ² we get λ² by itself on the right hand side. 2! = 2, so we just have 1/2 on the left hand side. 4/4! = 1/6, so we have λ²/6 on the right hand side.

1/2 = λ²/6

λ² = 3

λ = ±√3

However, λ can never be negative. So we know that, for a poisson random variable X, P(X = 2) = 4*P(X = 4) if and only if λ = √3.

We can check our answer in R:

> dpois(2,sqrt(3))

[1] 0.2653818

> dpois(4,sqrt(3))*4

[1] 0.2653818

Checks out.