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Conditional distribution

  1. Apr 22, 2007 #1
    I've been staring at this for hours. Any hints?

    Let the vector [tex]Y = (Y_1,Y_2,\dots,Y_k)[/tex] have a multinomial distribution with parameters n and [tex]\pi = (\pi_1,\pi_2,\dots,\pi_k)[/tex]:

    [tex]\sum_{i=1}^{k}Y_i = n, \quad \sum_{i=1}^{k}\pi_i = 1[/tex] ​

    Show that the conditional distribution of [tex]Y_1[/tex] given [tex]Y_1+Y_2=m[/tex] is binomial with n = m and [tex]\pi = \frac{\pi_1}{\pi_1+\pi_2}[/tex].

    I've tried to apply the definition of a conditional probability and sum over the relevant events in the multinomial distribution, but it gives me nothing.

    Thanks.
     
  2. jcsd
  3. Apr 22, 2007 #2
    Hmmm, this works if they are Poisson. Not sure if it works if it is multinomial. The standard way to do this would be to compute P[X_1=x|X_1+Y_2=m], as you tried.

    Edit: I should be more precise. If Y_1 and Y_2 poisson r.v. with paramaters lambda1 and lambda2, then Y_1 | Y_1+Y_2=m is distributed as Binomial(m, lambda1/(lambda1+lambda2)).
     
    Last edited: Apr 22, 2007
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