Results 1 to 4 of 4

Math Help - Help~ poisson distribution

  1. #1
    Newbie
    Joined
    Aug 2008
    Posts
    17

    Help~ poisson distribution

    a field cantains two types of plants. type A is attractive to insects, whereas B is not. The number of insects to be found on a plant of type A can be modelled as a poisson random variable with mean m. no insects will ever be found on a plant B. the proportion of type B plants in the field is p. let X be the number of insects found on a randomly chosen plant.

    1)find the probability mass function of X

    2)show that the probability generating function of X is given by
    π(s)=p+(1-p)*(e^(m*(s-1)). use this result to find the mean and variance of X.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Flow Master
    mr fantastic's Avatar
    Joined
    Dec 2007
    From
    Zeitgeist
    Posts
    16,948
    Thanks
    5
    Quote Originally Posted by kevek View Post
    a field cantains two types of plants. type A is attractive to insects, whereas B is not. The number of insects to be found on a plant of type A can be modelled as a poisson random variable with mean m. no insects will ever be found on a plant B. the proportion of type B plants in the field is p. let X be the number of insects found on a randomly chosen plant.

    1)find the probability mass function of X

    2)show that the probability generating function of X is given by
    π(s)=p+(1-p)*(e^(m*(s-1)). use this result to find the mean and variance of X.
    What have you tried so far? Where do you get stuck?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Aug 2008
    Posts
    17
    Quote Originally Posted by mr fantastic View Post
    What have you tried so far? Where do you get stuck?
    Hi, in first question, can i seperate pmf in two part (one with p, another with (1-p) )?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Flow Master
    mr fantastic's Avatar
    Joined
    Dec 2007
    From
    Zeitgeist
    Posts
    16,948
    Thanks
    5
    Quote Originally Posted by kevek View Post
    Hi, in first question, can i seperate pmf in two part (one with p, another with (1-p) )?
    The pmf of X is defined as follows:

    f(x) = (p)(1) + (1 - p)e^{-m} = p + (1-p) e^{-m} for X = 0

    (since X = 0 always for plant B and sometimes for Plant A),

    f(x) = (p)(0) + (1-p) \frac{m^x e^{-m}}{x!} for X > 0

    (since X is never greater than zero for Plant B but can be for Plant A).


    By definition n(s) = E(s^X)

     = \left[ p + (1-p) e^{-m}\right] s^0 + (1-p) \sum_{x=1}^{\infty} \frac{m^x e^{-m}}{x!} s^x

     = p + (1-p) e^{-m} + (1-p) \left[ \sum_{x=0}^{\infty} \frac{m^x e^{-m}}{x!} s^x - e^{-m} \right]

    The sum is just the probability generating function of a Poisson random variable and is well known (and is probably derived somewhere in these forums).


    You should know how to get mean and variance from a probability generating function.
    Last edited by mr fantastic; August 28th 2008 at 09:24 PM. Reason: Added the explanation in brackets
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 1
    Last Post: December 27th 2011, 02:08 PM
  2. Poisson distribution
    Posted in the Advanced Statistics Forum
    Replies: 4
    Last Post: June 5th 2011, 08:33 AM
  3. Poisson distribution and asymptotic normal distribution.
    Posted in the Advanced Statistics Forum
    Replies: 0
    Last Post: August 18th 2010, 06:52 AM
  4. Poisson Distribution
    Posted in the Advanced Statistics Forum
    Replies: 6
    Last Post: April 15th 2010, 03:30 AM
  5. Poisson Distribution
    Posted in the Advanced Statistics Forum
    Replies: 3
    Last Post: May 10th 2007, 12:41 AM

Search Tags


/mathhelpforum @mathhelpforum