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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?

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    What are your thoughts so far?2010-10-31
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    @Mike Spivey Well, i am thinking about trying to use exponential distribution which is linked with the poisson since it has the memoryless property but not sure if this is the right idea? or maybe i could use the fact that the first two probabilities are in different months so are independent (i think) and use the conditional probaility rule...what are you thinking?2010-10-31
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    Are you insisting that there are exactly 3 claims in the first two months AND (or OR) 5 claims in the first 3 months (given that there are five in the first four months)?2010-11-01

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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?

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    It looks great. I particularly like that you explicitly worked the Poisson probabilities in there. (In fact, if this is a textbook problem, I suspect that you worked it the way it is "supposed to be" worked.)2010-11-01
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    The fancy formatting uses LaTeX. See http://meta.math.stackexchange.com/questions/934/where-is-the-latex-reference-please2010-11-01
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(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:

  1. Exactly 3 claims in the first two months.
  2. Exactly 2 claims in the third month.
  3. 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 $$\frac{5!}{3! 2! 0!} \left(\frac{1}{2}\right)^3 \left(\frac{1}{4}\right)^2 \left(\frac{1}{2}\right)^0 = \frac{5}{64}.$$

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    i am confused...i understand why we need to consider binomial distributions but i dont know how to calculate the probabilites for each distribution. I mean i dont know what to calculate. could you enlighten me?2010-11-01
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    so for example the first distribution, binomial with n = 5 and p=q=0.5, what do i use to calculate what probability? should i use tables? if so which value do i need to take as normally a binomial distribution has a parameter x? its been a long time since i used binomial distribution2010-11-01