claims in an insurance portfolio Claims arrive in an insurance portfolio, according to a homogenous Poisson process $X(t), t\geq 0$. Assume that each of the $12$ months in a year has exactly $30$ days. Calculate the value of the following conditional probability:
\begin{equation*}
P\{X(2) = 3, X(3) = 5 | X(4) = 5\}
\end{equation*}
where $X(k)$ is the number of claims during the first $k$ months of the year.
Can anyone help?
 A: (Major rewrite: The following is a much simpler and cleaner answer than my previous one involving binomial distributions.)
Start by considering what $X(4) = 5$ actually means.  It means that through the first four months of the year there have been 5 claims.  Given that, you're looking for the probability that all three of the following events occurred:


*

*Exactly 3 claims in the first two months.  

*Exactly 2 claims in the third month.  

*Exactly 0 claims in the fourth month.  


Each claim is equally likely to fall in each month.  So, out of 5 total items, you're trying to fit 3 into a category with a probability of $\frac{1}{2}$, 2 into a category with a probability $\frac{1}{4}$, and 0 into a category with probability $\frac{1}{4}$.  This situation is exactly what a multinomial distribution models.
So the probability is 
$$\frac{5!}{3! 2! 0!} \left(\frac{1}{2}\right)^3 \left(\frac{1}{4}\right)^2 \left(\frac{1}{2}\right)^0 = \frac{5}{64}.$$
A: thanks for your re-edit...now it makes much more sense and you are right, much cleaner!  
Since I was last working on the problem, I have come up with the following alternative solution which yields the same numerical answer as you:
P{X(2) = 3, X(3) = 5 | X(4) = 5}
= P{X(2) = 3, X(3) = 5, X(4) = 5} / P{X(4) = 5}
= P{X(2) = 3, X(3) - X(2) = 2, X(4) - X(3) = 0} / P{X(4) = 5} = (using independent increments)
= P{X(2) =3}.P{X(3) - X(2) = 2}.P{X(4) - X(3) = 0} / P{X(4) = 5
= {exp((1/3!)-2u(2u)^3).exp((1/2!)-u(u)^2).exp(-u)} / exp((1 / 5!)-4u(4u)^5)
= 5!(2)^3 / 3!2!1!(4)^5
= (4 . 5 . 5) / 4^5
= 5/64
what do you think of this approach?
also, sorry if it looks messy everyone, i just dont know how to get fancy formatting using the text entry, can anyone enlighten me on this?
