Results 1 to 2 of 2

Math Help - Poisson problem - Help!

  1. #1
    Super Member Deadstar's Avatar
    Joined
    Oct 2007
    Posts
    722

    Poisson problem - Help!

    Poisson question here thats causing me problems...

    A mountain hut is available for walkers to stay overnight. Members of the organisation which owns the hut have keys; others can turn up in the hope a keyholder will let them in. So, on any given night, if at least one keyholder turns up, everyone who turns up gets in; otherwise no-one gets in.

    Suppose that keyholders arrive randomly at a hut, so that on a given night the number arriving follows a Poisson distribution with mean λ. Suppose also that the number of non-keyholders arriving follows a Poisson distribution with mean , and is independent of the keyholders. Let X be the number of people occupying the hut (on a given night). Write down an expression for the probability that k keyholders and l non-keyholders occupy the hut, where k > 0, and by doing an appropriate summation show that

    P(X = n) = [(λ + )^n - ^n . e^-λ-]/n! (sorry dont have any proper math symbol program.)

    Consider the situation described above, with λ = 3 and = 5

    1) On average, how many people are expected to be in the hut per night?
    2) On a given night, 4 people are in the hut. What is the probability that they are all keyholders?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    Quote Originally Posted by Deadstar View Post
    Poisson question here thats causing me problems...

    A mountain hut is available for walkers to stay overnight. Members of the organisation which owns the hut have keys; others can turn up in the hope a keyholder will let them in. So, on any given night, if at least one keyholder turns up, everyone who turns up gets in; otherwise no-one gets in.

    Suppose that keyholders arrive randomly at a hut, so that on a given night the number arriving follows a Poisson distribution with mean λ. Suppose also that the number of non-keyholders arriving follows a Poisson distribution with mean , and is independent of the keyholders. Let X be the number of people occupying the hut (on a given night). Write down an expression for the probability that k keyholders and l non-keyholders occupy the hut, where k > 0, and by doing an appropriate summation show that

    P(X = n) = [(λ + )^n - ^n . e^-λ-]/n! (sorry dont have any proper math symbol program.)

    Consider the situation described above, with λ = 3 and = 5

    1) On average, how many people are expected to be in the hut per night?
    2) On a given night, 4 people are in the hut. What is the probability that they are all keyholders?
     <br />
p(k,l)=\frac{\lambda^k e^{-\lambda}}{k!} ~\frac{\mu^l e^{-\mu}}{l!}<br />

     <br />
P(X=n)=\sum_{k=1}^n p(k,n-l)<br />
=\sum_{k=1}^n \frac{\lambda^k e^{-\lambda}}{k!} ~\frac{\mu^{n-k} e^{-\mu}}{(n-k)!}<br />
<br />
=\frac{e^{-\lambda-\mu}}{n!}\sum_{k=1}^n \frac{n!}{k!(n-k)! }\lambda^k \mu^{n-k}<br />
<br />
=\frac{e^{-\lambda-\mu}}{n!} \left( (\lambda + \mu)^n-\mu^n\right)<br />

    The last step uses a rearrangement of the binomial theorem.

    RonL
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Problem with Poisson
    Posted in the Advanced Statistics Forum
    Replies: 2
    Last Post: November 1st 2010, 02:05 PM
  2. Poisson Square Wave (PSW) and Filtered Poisson Process (FP)
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: April 5th 2010, 12:56 PM
  3. Replies: 0
    Last Post: October 8th 2009, 08:45 AM
  4. Poisson Problem
    Posted in the Advanced Statistics Forum
    Replies: 3
    Last Post: January 10th 2009, 12:07 AM
  5. Poisson Problem
    Posted in the Advanced Statistics Forum
    Replies: 5
    Last Post: January 8th 2009, 01:29 AM

Search Tags


/mathhelpforum @mathhelpforum