Probability question; Conditional probability and poisson distribution

  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. fzero

    fzero 3,005
    Science Advisor
    Homework Helper
    Gold Member

    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...
Know someone interested in this topic? Share this thead via email, Google+, Twitter, or Facebook

Have something to add?
Similar discussions for: Probability question; Conditional probability and poisson distribution