# Sickness occurrence distribution among populations and poisson random variable

The number of times someone gets sick in a year is given by a Poisson random variable with parameter $$\lambda = 5$$. Let us suppose there is a new medicine which reduces the parameter lambda to $$\lambda = 3$$ for $$75\%$$ of the population. For the remaining $$25\%$$, the drug has no effect. If a person takes the medicine for a whole year and gets sick twice, what is the probability that the drug is effective to him?

MY ATTEMPT

Let us divide the population into two parts according to the parameter lambda: $$P_{1}$$ and $$P_{2}$$. Moreover, let us also denote by $$N$$ the number of times the person has got sick in the year. Therefore we are interested in the probability:

\begin{align*} \textbf{P}(P_{1}\mid N = 2) = \frac{\textbf{P}(P_{1}\cap\{N = 2\})}{\textbf{P}(N = 2)} = \frac{\textbf{P}(N = 2\mid P_{1})\textbf{P}(P_{1})}{\textbf{P}(N = 2\mid P_{1})\textbf{P}(P_{1}) + \textbf{P}(N = 2\mid P_{2})\textbf{P}(P_{2})} \end{align*}

where

$$\begin{cases} \textbf{P}(P_{1}) = 0.75\\\\ \textbf{P}(P_{2}) = 0.25\\\\ \textbf{P}(N = 2\mid P_{1}) = \displaystyle\frac{e^{-3}\times 3^{2}}{2!}\\\\ \textbf{P}(N = 2\mid P_{2}) = \displaystyle\frac{e^{-5}\times 5^{2}}{2!} \end{cases}$$

I just would like that someone could double-check my solution and correct it if necessary. Thanks in advance!

• You haven't answered the final question. – herb steinberg Jan 26 at 22:27
• Unlesss I am distracted, the answer is given by $\textbf{P}(P_{1}\mid N = 2)$, whose expression I have determined. I was just wondering if my reasoning is right. Anyway, I apologize if this is not the case. – user1337 Jan 26 at 22:38
• Your reasoning is correct. I missed the fact that you had stated it at the beginning, since you didn't put it all together at the end. – herb steinberg Jan 26 at 23:08
• No problems. Thanks for the contribution, anyway. – user1337 Jan 26 at 23:31