Question here:
Iwana Passe is taking a multiple-choice exam. You may assume that the number of questions is infinite. Simultaneously, but independently, her conscious and subconscious faculties are generating answers for her, each in a Poisson manner. (Her conscious andsubconscious are always working on different questions.) Conscious responses are generated at the rate $λ_c$ responses per minute. Subconscious responses are generated at the rate $λ_s$ responses per minute. Assume $λ_c \neq λ_s$. Each conscious response is an independent Bernoulli trial with probability $p_c$ of being correct. Similarly, each subconscious response is an independent Bernoulli trial with probability $p_s$ of being correct. Iwana responds only once to each question, and you can assume that her time for recording these conscious and subconscious responses is negligible.
All my solutions are based on the fact that conscious and subconscious responses form the merged Poisson process.
(a) If we pick an interval of T minutes, what is the probability that in that interval Iwana will make exactly $r$ conscious responses and $s$ subconscious responses?
My solution: Because this is the merged Poisson process, so the probability when conscious happens is $\frac{\lambda_c}{\lambda_c+\lambda_s} $ , similar for the subconscious. Thus the probability is $$ \binom{r+s}{r} \frac{(\lambda_c)^r(\lambda_s)^s}{(\lambda_c+\lambda_s)^{s+r}} $$
But the given answer is:
Since the conscious and subconscious responses are generated independently,
P($r$ conscious responses and s subconscious responses in interval $T$)
= P($r$ conscious responses in $T$)P($s$ unconscious responses in $T$) = $$ \frac{(\lambda_cT)^re^{-\lambda_cT}}{r!}\cdot \frac{(\lambda_sT)^se^{-\lambda_sT}}{s!}$$
I think that the total events are given, the interval T is not in the business.
Appreciate if you help! Thanks!
A similar question:
What you wrote is $\textsf{Pr}(R=r\land S=s\mid R+S=r+s)$. What they wrote (and what the question asked for) is $\textsf{Pr}(R=r\land S=s)$. The latter depends on the interval, the former doesn't.