Results 1 to 3 of 3

Math Help - Large Poisson Distribution

  1. #1
    Newbie
    Joined
    Feb 2014
    From
    Wales
    Posts
    1

    Large Poisson Distribution

    I need to find the probability of 320 or more flaws in a 10km length of fibre optic cable. There is an average of 1.5 flaws per 50m.

    I could solve this using:

    P(X≥320) = 1 - [P(X=0) + P(X=1) + ... + P(X=319)]

    But this would be extremely long and tedious to do, plus it wouldn't be an appropriate method.

    Is there any other way I could get to my answer in a appropriate method?
    (post links if necessary)
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Nov 2013
    From
    California
    Posts
    2,690
    Thanks
    1086

    Re: Large Poisson Distribution

    Quote Originally Posted by Foflo View Post
    I need to find the probability of 320 or more flaws in a 10km length of fibre optic cable. There is an average of 1.5 flaws per 50m.

    I could solve this using:

    P(X≥320) = 1 - [P(X=0) + P(X=1) + ... + P(X=319)]

    But this would be extremely long and tedious to do, plus it wouldn't be an appropriate method.

    Is there any other way I could get to my answer in a appropriate method?
    (post links if necessary)
    Wiki shows a closed form CDF for the Poisson distribution if you can numerically calculate the incomplete Gamma function

    $$F_{\lambda}[k]=\frac{\Gamma\left(\lfloor k+1 \rfloor, \lambda\right)}{\lfloor k \rfloor !}$$
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor

    Joined
    Apr 2005
    Posts
    16,057
    Thanks
    1690

    Re: Large Poisson Distribution

    An average of 1.5 flaw per 50m is 3 flaws per 100 m so 30 flaws per km and 300 flaws per 10 km.
    With a Poisson distribution with parameter [itex]\lambda=300[/itex] has mean and standard deviation [itex]300[/itex] so, like any distribution with finite mean and standard deviation, can be approximated by a normal distribution with that same mean and standard deviation. That is, we can approximate this by the normal distribution with mean and standard deviation 300. The probability x is greater than 320 can be changed to the probability that x is greater than 319.5 (since the normal distribution is continuous, we use the "half integer correction" treating any number from 319.5 to 320.5 as "320"). That converts to the standard normal distribution z\ge (319.5- 300)/300= 0.065. Now look it up in a table of the normal distribution, such as the one at http://www.stat.duke.edu/~banks/111-...mDistTable.pdf. Since that gives the probability a value is less than or equal to z, subtract the value in the table from 1.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 2
    Last Post: September 22nd 2012, 09:03 AM
  2. Replies: 1
    Last Post: December 27th 2011, 01:08 PM
  3. Poisson distribution and asymptotic normal distribution.
    Posted in the Advanced Statistics Forum
    Replies: 0
    Last Post: August 18th 2010, 05:52 AM
  4. Sum of Geometric distribution for large n
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: July 16th 2010, 05:15 PM
  5. Poisson approximated by Normal for large numbers?
    Posted in the Statistics Forum
    Replies: 1
    Last Post: November 18th 2008, 12:38 PM

Search Tags


/mathhelpforum @mathhelpforum