# Thread: Proof about independence of a number of Poisson random variables

1. ## Proof about independence of a number of Poisson random variables

Suppose that the number of events occuring in a given time period is a Poisson random variable with parameter λ. If each event is classified as a type i event with probability $p_i$, i = 1,...,n, sum of $p_i$ = 1, independently of other events, show that the numbers of type i events that occur, i = 1,...,n, are independant Poisson random variables with parameters λ $p_i$, i = 1,...,n.

I am finding this very unclear and confusing. or maybe I'm jsut burnt out from the rest of my studying

Anyone have any advice/ideas that could help clear this up.

2. What part are you getting stuck on?

3. I'm not clear on what is being said

4. Well, they tell you that a Random Variable has the Poisson distribution with lambda. Each event, is an independent Poisson random variable (since it is impossible to observe 5 and say 7 in the same event yes?). Your job is to show that the mean of the independent random variable is equal to lambda times the probability of event "i" (since the mean is the parameter of a Poission random variable).

Hopefully I didn't just restate something you already knew.

5. ok ' I'll let you know in a bit how it goes sorry i took a while to reply.. was taking some of my weekend off from math :P

6. ok this does make sense to me now... but this question is turning out to be insane

i used the multinomial distribution (a sequence of n independent and identical experiments) and it gives me some complicated nonesense with very crappy factorials.. i can see almost that this could lead me in the right direction.. but nope... no luck.

from the multinomial thing i ended up with

(P{X_i=i})/(n-i+1) = n/i!

in the end... no real factorials left... but... well this isnt telling me anything.

i just did a bunch of factoring to get this... but something tells me this is a crappy direction to go

7. ahaha! i cracked it... sadly its too long to really type up.

i used the multinomial dist and set it up as a conditional probability times the probability of the given which was the sum of the ns

and a lot of factoring later i got a poisson process multiplied by r-1 other poisson processes with pilambda as their parameter. yay!