So I'm told that claims arrive in an insurance portfolio to a homogeneous Poisson process with at a rate of $\lambda > 0$ . Let $X(t), t\geq 0$ denote the number of claims that arrive in the time interval $(0,t]$ and let $P_k(t)=P(X(t)=k), \ k=0,1,2,...$
I'm asked to verify that:
- $P(X(t)$ is odd$)=((e^{\lambda t} - e^{-\lambda t})/2)e^{-\lambda t}$
- $P(X(t)$ is even$)= ((e^{\lambda t}+e^{-\lambda t})/2)e^{-\lambda t}$
I'm not really sure at all what I need to do. I know how to answer questions that ask to find the probability of a certain number of claims in a given time period (such as October to December) but I'm not sure how I can apply that here.
The only idea I can think of is using the general formula: $P(X=k)=(\lambda^k/k!)e^{-\lambda}$ where $\lambda$ is the rate of claims and $k$ is the number of claims, maybe I could pair this with the idea that an even number of claims is $2k$ and odd number is $2k-1$?
I also know the probability formula for a non-homogeneous Poisson Process: say we want to find the probability of $k$ claims from $t_1$ to $t_2$ where $t_1,t_2$ are months in the year, then the probability formulas would be: $e^{-(\Lambda(t_2)-\Lambda(t_1))}\cdot (\Lambda (t_2)-\Lambda (t_1)^k)/k!$ which looks quite similar to what they are asking me to verify, but again I'm not sure how I can use this.
Both of these ideas are just guesses so feel free to ignore them if I'm completely off :)
Any help would be greatly appreciated
Thanks in advance