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Probability Mass Functions of Binomial Variables

  1. Oct 12, 2009 #1
    1. The problem statement, all variables and given/known data
    Let X and Y be independent binomial random variables with parameters n and p.
    Find the PMF of X+Y.
    Find the conditional PMF of X given that X+Y=m.


    2. Relevant equations
    The PMF of X is P(X=k)=(n C k)pk(1-p)n-k
    The PMF for Y would be the same.

    3. The attempt at a solution
    I am really not sure how to go about solving this problem though I have been told that the first part can be done with no computation or calculations at all. Not sure at all on the second part.
     
  2. jcsd
  3. Oct 12, 2009 #2

    jbunniii

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    Do you know anything in general about the PMF of the sum of two independent discrete random variables? Does "convolution" ring a bell?

    If not, don't worry about it. Think about what a binomial distribution with parameters n and p means. You can think of it as the number of "heads" resulting from flipping a coin n times, where the probability of "head" is p. If you do that experiment TWICE, independently, and the results are X and Y, respectively, then X + Y is equivalent to simply flipping the coin 2n times, right? What's the PMF for that experiment?
     
    Last edited: Oct 12, 2009
  4. Oct 12, 2009 #3
    Hmm, not that I know of.
    I haven't heard the term convolution before.
    Could you provide me with a definition?
     
  5. Oct 12, 2009 #4

    jbunniii

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    You need it to answer the question in general (for arbitrary PMFs), but don't worry about it in this case - for this particular example you can solve it another way (see my edited post above).
     
  6. Oct 12, 2009 #5
    Ah yes, I understand it now.
    Thanks very much for your clear explanation.
     
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