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?


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*}


\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!

  • $\begingroup$ You haven't answered the final question. $\endgroup$ – herb steinberg Jan 26 at 22:27
  • $\begingroup$ 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. $\endgroup$ – user1337 Jan 26 at 22:38
  • $\begingroup$ 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. $\endgroup$ – herb steinberg Jan 26 at 23:08
  • $\begingroup$ No problems. Thanks for the contribution, anyway. $\endgroup$ – user1337 Jan 26 at 23:31

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