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What is the expectation of the number of great-grandsons a cell have?

  1. Sep 25, 2008 #1
    1. The problem statement, all variables and given/known data

    A cell diverges into X new cells. Each of them reproduces in the same manner. X is a geometric random variable with success parameter of 0.25.
    What is the expectation of the number of great-grandsons a cell have?

    2. The attempt at a solution
    I thought about using the formula EX=EEX|Y somehow, but it wasn't very useful:
    If the number of children is C
    and the number of grandsons is G
    and the number of great-grandsons is N
    then
    N|G=GX
    G|C=CX
    C=X

    here I got stuck, is it the right direction at all?
    thanks..:)
     
  2. jcsd
  3. Sep 26, 2008 #2
    Re: probability

    The answer is simpler than that. This is an example of a Galton-Watson type of branching process.
     
  4. Sep 26, 2008 #3
    Re: probability

    thanks

    but I've never heard of this process, not in our lectures, at least...
    is there any other way?
     
  5. Sep 26, 2008 #4
    Re: probability

    Let [tex]Z_n[/tex] be the number of offspring for generation n. Prove the recursion relation [tex]E[Z_n] = E[Z_1]E[Z_{n-1}][/tex], where [tex]Z_0=1[/tex] is the first generation. (Great-grandsons occurs at generation n=3.) You know that [tex]Z_1[/tex] is a geometrically distributed random variable as given and can find [tex]E[Z_1][/tex]. Now, note that the expected value of the number of offspring at generation n-1 will be the same as the expected value of the number of offspring that each "son" has at generation n. (It helps to draw a family tree.) Therefore, the number of offspring [tex]Z_n[/tex] is a sum of [tex]Z_1[/tex] random variables each with probability distribution of [tex]Z_{n-1}[/tex]. You should now be able to prove the recursion.
     
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