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Math Help - Poisson distribution

  1. #1
    Senior Member chella182's Avatar
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    Poisson distribution

    It's only the last two parts of this I'm having a bit of bother with, but I'll type the whole question out so you know what's going on.

    Suppose that the number of times during a year that an individual catches a cold can be modelled by a Poisson random variable with an expectation of 4. Further suppose that a new drug based on Vitamin C reduces the expectation to 2 (but is still a Poisson distribution) for 80% of the population, but has no effect on the remaining 20% of the population. Calculate

    a) the probability that an individual taking the drug has 2 colds in a year if they are part of the population which benefits from the drug;

    b) the probability that an individual has 2 colds in a year if they are part of the population which does not benefit from the drug;

    c) the probability that a randomly chosen indicidual has 2 colds in a year if they take the drug;

    d) the conditional probability that a randomly chosen individual is in the part of the population which benefits from the drug given that they had 2 colds in a year during which they took the drug.

    Pretty lengthy, sorry.
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  2. #2
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    Quote Originally Posted by chella182 View Post
    It's only the last two parts of this I'm having a bit of bother with, but I'll type the whole question out so you know what's going on.

    Suppose that the number of times during a year that an individual catches a cold can be modelled by a Poisson random variable with an expectation of 4. Further suppose that a new drug based on Vitamin C reduces the expectation to 2 (but is still a Poisson distribution) for 80% of the population, but has no effect on the remaining 20% of the population. Calculate

    a) the probability that an individual taking the drug has 2 colds in a year if they are part of the population which benefits from the drug;

    b) the probability that an individual has 2 colds in a year if they are part of the population which does not benefit from the drug;

    c) the probability that a randomly chosen indicidual has 2 colds in a year if they take the drug;

    d) the conditional probability that a randomly chosen individual is in the part of the population which benefits from the drug given that they had 2 colds in a year during which they took the drug.

    Pretty lengthy, sorry.
    I've seen longer, and the context is never bad to have, so thanks!

    c) hint: \mathbf{P}(N=2) = \mathbf{P}((N=2 \textrm{ and they benefit}) \textrm{ or } (N=2 \textrm{ and they do not benefit}))

    break out "or", note the two sets are disjoint so the intersection is empty
     = \mathbf{P}(N=2 \textrm{ and they benefit}) +   \mathbf{P}(N=2 \textrm{ and they do not benefit}) - 0

     =  \mathbf{P}(N=2 |\textrm{they benefit})\mathbf{P}(\textrm{they benefit}) + \ldots

    d)  \mathbf{P}(\textrm{they benfit}|N=2) =   \mathbf{P}(\textrm{they benefit and }N=2)/\mathbf{P}(N=2)

    You had 2 figure out what both the numerator and denominator were in the previous part - if you followed my hint
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  3. #3
    Senior Member chella182's Avatar
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    Thanking you muchly looking at my answers for part a) and b) I'm not sure they're right, but still haha.
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  4. #4
    Senior Member chella182's Avatar
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    Poisson distribution problem

    Okay, so I've looked at my answers to parts a) and b) to the question in the first post & I don't think they're right. Can anyone help?
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