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Conditional Multinomial Problem

  1. Jun 1, 2010 #1
    If Y1,Y2,Y2 ~ Multinomial with parameter (n,p1,p2,p3)
    Prove that the conditional distribution of Y1 given Y3=m (m<n)
    is a binomial with ( (n-m), p1/(p1+p2) )

    p1+p2+p3=1
    y1+y2+y3=n
    y3=m

    My Attempt:
    P( Y1=y1| y3=m) = P(Y1=y1, Y3=m)/ P(Y3=m)

    ( n choose m and y1 ) p1^y1*p3^m / ( n choose m) p3^m*(p1+p2)^n-m

    leaving me with
    (n-m)!/ y1! (p1/ p1+p2)^y1((p1+p2) ^y2))
    ....
    can't seem to simplyfy this to become a binomial
    honestly stuck here!

    Thanks!
     
  2. jcsd
  3. Jun 7, 2010 #2

    EnumaElish

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    (n choose m)(n-m choose y1) p1^y1*p2^(n-m-y1)*p3^m
    divided by
    (n choose m) p3^m*(p1+p2)^(n-m)

    simplifies to

    (n-m choose y1)p1^y1 p2^(n-m-y1) (p1 + p2)^(m - n). Rearrange to obtain the result.
     
    Last edited: Jun 8, 2010
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