1. ## 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.

2. Originally Posted by chella182
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

3. Thanking you muchly looking at my answers for part a) and b) I'm not sure they're right, but still haha.

4. ## 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?