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Probability question; Conditional probability and poisson distribution

  1. Nov 1, 2010 #1
    1. The problem statement, all variables and given/known data

    A radioactive source emits particles according to a Poisson process, at an average rate of λ per unit time. Each particle emitted has probability p of being detected by an instrument, independently of other particles. Let X be the number of particlese emitted in a given time interval of length T , and Y the number of those particles that are detected. As usual, let μ = λT and q = 1 − p.
    (i) What is the conditional probability p(Y = r|X = k)?

    2. Relevant equations

    I know that I want p(Y=r|X = k) = p((Y=r) ∩ (X=k)) / p(X=k)

    I know from poisson distribution that p(X=k) = ((μ^k)/k!)*e^(−μ)

    3. The attempt at a solution

    I don't understand how I can work out what p((Y=r) ∩ (X=k)) equals but this is my attempted solution:

    Since there are k particles emitted and we want to know the probability that r of them have been detected then using binomial distribution:

    p((Y=r) ∩ (X=k)) = [k choose r](p^r)(q^(k-r))

    I know this is wrong but I can't quite work out how to tie it all together...
  2. jcsd
  3. Nov 2, 2010 #2


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    You wrote down a formula assuming that k particles have been emitted. Therefore this is not the joint probability, but it is still relevant to the conditional probability that you want to compute...
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