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Homework Help: Probability - Conditional Expectation

  1. Apr 23, 2012 #1
    My professor explained this concept absolutely horribly and I have no idea how to do these problems.

    Let A and B be independent Poisson random variables with parameters α and β, respectively. Find the conditional expectation of A given A + B = c.
    (Hint: For discrete random variables, there is no conditional density. Use the definition of conditional probability.)

    Starting with the definition, f(A | A + B = c) = [f(A, A+B=c)] / [f(A+B=c)]

    Not sure how to proceed.
  2. jcsd
  3. Apr 23, 2012 #2
    We will have to work with A+B. Do you know the probability distribution of this??
  4. Apr 23, 2012 #3

    Ray Vickson

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    It is best to be clear and to use correct notation: you want P{A=k|A+B=c} for the possible values of k in {0,1,2,...}. So, you need to compute P{A=k & A+B=c} in the numerator (and, of course, you need P{A+B=c} in the denominator).

    Do you know the distribution of A+B? It should be in your textbook or course notes; if not, look on-line, or work it out for yourself from first principles, using the distributions of A and B and the formula for the distribution of a sum of independent random variables (really: it is not that hard!).

    Last edited: Apr 23, 2012
  5. Apr 23, 2012 #4
    The distribution for a Poisson distribution is p(x) = [e^(-λ)*λ^x] / x!
  6. Apr 23, 2012 #5
    Yes, that is the distribution for A and B with [itex]\lambda=\alpha[/itex] and [itex]\lambda=\beta[/itex] respectively.

    But we are asking about the distribution of A+B.
  7. Apr 24, 2012 #6
    A + B also has a Poisson distribution with parameters Poisson(A+B), as the sum of independent Poisson random variables has a Poisson distribution.
  8. Apr 24, 2012 #7
    OK, that's good. Now we want to figure out (for fixed c)


    In order to find to, we want to find


    Of course, this is equal to


    Can you find this?? This is just a two-dimensional pmf. Remember that A and B are independent, so you can find it easily.

    Then we also nee to find


    This should be easy since you just figured out the distribution of A+B.
  9. Apr 24, 2012 #8
    Not sure how to go about finding f(A=x , B=c−x)
  10. Apr 24, 2012 #9

    Ray Vickson

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    You told us that A and B are independent. What do you think that means?

  11. Apr 25, 2012 #10
    Can anyone show me this problem step by step? I'm not picking up on any of this question, which is why I posted this.
  12. Apr 25, 2012 #11

    Ray Vickson

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    No, we can't. That is not how this forum works. However, I will give you a hint: if your professor did not explain things to your satisfaction, and if, for some reason you do not have access to course notes or to a textbook, then *look online*. Google 'independent + probability' to turn up hundreds of articles at various levels of sophistication, from step-by-step explanations to abstract discussions.

    Last edited: Apr 25, 2012
  13. Apr 25, 2012 #12
    = P(A|B=c-A)
    = P(A and B=c-A) / P(B=c-A)

    =P(α + β) / P(β) ?
  14. Apr 25, 2012 #13

    Ray Vickson

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    I have no idea what you mean by P(α + β) or P(β). I know what α and β are, and I know what is meant by P(A=u) or P(B=v) and how to write them in terms of α, β, u and v, but I cannot figure out your P(α+β), etc. Anyway, I certainly would get something very different from what you wrote.

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