Probability of claims using homogeneous Poisson distribution

I'm told that claims arrive at an insurance portfolio according to a homogeneous Poisson process where the mean waiting time until the first claim is $2$ months. Assume all $12$ months have $30$ days.

I want to find the probability that $2$ claims arrive in April and $1$ claim arrives in July. I know this is relatively straightforward but since I haven't practiced this in a while I've forgotten all the steps can could use some help.

Here's what I've done:

First we find the average claims a month $\lambda$, in this case it is $1/2$ claims/month (since, on average, we have $1$ claim arrive in $2$ months).

Next I wrote out the probabilities we want to find: $P${$X(4)-X(3)=2,X(7)-X(6)=1$} (since April is the $4th$ month and July is the $7th$). Using homogeneity we can re-write this as: $P${$X(4)-X(3)=2$}$\cdot P${$X(7)-X(6)=1$}

From here I've forgotten the formula I use to get my answer, I think the formula for $P${$X(4)-X(3)=2$} is: $((1/2)^2e^{-1/2})/2!$ but this might be wrong.

The answer on the solution sheet is $0.2229$. (No working out is provided hence why I'm not sure what to do, also some other answers on the solution sheet have been incorrect for other questions so this may not be the correct answer but it is likely to be this.)

If anyone could show me the method I'd use to get my answer once I've written out the probabilities it would really help.

Thanks in advance

Solution 1:

What you have done is correct.

$ \displaystyle \small P(2 \text{ claims in April}) = ~\frac{(1/2)^2}{2!} e^{-1/2} = \frac{1}{8 \sqrt e}$

$ \displaystyle \small P(1 \text{ claim in July}) = ~\frac{1}{2 \sqrt e}$

Given you have two non-overlapping intervals, you can write

$\displaystyle \small P(2 \text{ claims in April and } 1 \text{ claim in July})$
$\displaystyle \small = P(2 \text{ claims in April}) \cdot P(1 \text{ claim in July}) = \frac{1}{16e} \approx 0.02299$

So the given answer seems to have decimal in the wrong place.