Results 1 to 4 of 4

Math Help - Hypergeometric Probability

  1. #1
    Junior Member
    Joined
    Jul 2006
    Posts
    43

    Hypergeometric Probability

    Have a problem:

    The sizes of animal populations are often estimated by using a capture-tag-recapture method. In this method k animals are captured, tagged, and then released into the population. Some time later n animals are captured, and Y, the number of tagged animals among the n, is noted. The probabilities associated with Y are a function of N, the number of animals in the population, so the observed value of Y contains information on this unknown N. Suppose that k=4 animals are tagged and then released. A sample of n=3 animals is then selected at random from the same population. Find P(Y=1) as a function of N. What value of N will maximize P(Y=1).

    Note that for the following [a,b] is the combinations rule such that this equals a!/b!(a-b)!.

    I said to let X be the number of tagged animals among the n. X therefore is a hypergeometric probability distribution with r=4, y=1, n=3, and N=N. Therefore the distribution would be ([4,1]*[N-4,2])/[N,3].

    Is this right and is this a function of N. How would you maximize this function?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member
    Joined
    Apr 2006
    Posts
    399
    Awards
    1
    Quote Originally Posted by JaysFan31 View Post
    Have a problem:

    The sizes of animal populations are often estimated by using a capture-tag-recapture method. In this method k animals are captured, tagged, and then released into the population. Some time later n animals are captured, and Y, the number of tagged animals among the n, is noted. The probabilities associated with Y are a function of N, the number of animals in the population, so the observed value of Y contains information on this unknown N. Suppose that k=4 animals are tagged and then released. A sample of n=3 animals is then selected at random from the same population. Find P(Y=1) as a function of N. What value of N will maximize P(Y=1).

    Note that for the following [a,b] is the combinations rule such that this equals a!/b!(a-b)!.

    I said to let X be the number of tagged animals among the n. X therefore is a hypergeometric probability distribution with r=4, y=1, n=3, and N=N. Therefore the distribution would be ([4,1]*[N-4,2])/[N,3].

    Is this right and is this a function of N. How would you maximize this function?
    This looks right to me. ([4,1]*[N-4,2])/[N,3] is P(Y=1), not the distribution. Writing out this probability as factorials, cancelling where possible, and ignoring any factor not having an N in it, I get P(Y=1) ~ (N-4)(N-5)/N(N-1)(N-2). This isn't nice to maximize using differentiation, so I plugged it into a spreadsheet and tried values. I found N = 10 was the maximizer. I did this in a hurry, so please confirm my calculations.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Jul 2006
    Posts
    43
    I got 12((N-4)(N-5)/N(N-1)(N-2)). I found the maximum of this to be 3. Can anyone confirm this?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Senior Member
    Joined
    Apr 2006
    Posts
    399
    Awards
    1
    Quote Originally Posted by JaysFan31 View Post
    I got 12((N-4)(N-5)/N(N-1)(N-2)). I found the maximum of this to be 3. Can anyone confirm this?
    The value 3 cannot be either the maximizer N or the function value at the maximizer.

    First, the formula would be invalid at N = 3 as it is not possible to take a sample without replacement of size 4 from a population of size 3.

    Second, the formula is for a probability so its value is between 0 and 1.
    Last edited by JakeD; September 26th 2006 at 07:57 PM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Hypergeometric
    Posted in the Advanced Statistics Forum
    Replies: 4
    Last Post: March 22nd 2011, 12:46 PM
  2. Hypergeometric problem
    Posted in the Advanced Statistics Forum
    Replies: 0
    Last Post: November 3rd 2009, 06:16 PM
  3. Probability - Hypergeometric random variable
    Posted in the Advanced Statistics Forum
    Replies: 5
    Last Post: March 1st 2009, 10:26 AM
  4. hypergeometric question
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: March 15th 2008, 11:43 PM
  5. I need help about hypergeometric series
    Posted in the Calculus Forum
    Replies: 6
    Last Post: April 16th 2006, 11:15 AM

Search Tags


/mathhelpforum @mathhelpforum