Results 1 to 7 of 7

Math Help - Proof about independence of a number of Poisson random variables

  1. #1
    Junior Member
    Joined
    Mar 2009
    Posts
    33

    Unhappy 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.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member
    Joined
    Jul 2009
    Posts
    593
    Thanks
    4
    What part are you getting stuck on?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Mar 2009
    Posts
    33
    I'm not clear on what is being said
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Super Member
    Joined
    Jul 2009
    Posts
    593
    Thanks
    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.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Junior Member
    Joined
    Mar 2009
    Posts
    33
    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
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Junior Member
    Joined
    Mar 2009
    Posts
    33
    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 isn`t telling me anything.

    i just did a bunch of factoring to get this... but something tells me this is a crappy direction to go
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Junior Member
    Joined
    Mar 2009
    Posts
    33
    ahaha! i cracked it... sadly it`s 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!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Independence of random variables X and X^2
    Posted in the Statistics Forum
    Replies: 2
    Last Post: December 9th 2011, 03:22 AM
  2. Independence of Random Variables
    Posted in the Statistics Forum
    Replies: 1
    Last Post: July 18th 2011, 07:39 PM
  3. Proof of independence of random variables
    Posted in the Advanced Statistics Forum
    Replies: 10
    Last Post: September 28th 2009, 12:31 PM
  4. stochastic independence and random variables
    Posted in the Advanced Statistics Forum
    Replies: 2
    Last Post: November 4th 2008, 11:12 PM
  5. 3 Gaussian random variables, independence
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: May 12th 2007, 04:06 AM

Search Tags


/mathhelpforum @mathhelpforum