This question came up while reading a medical paper - the study showed $m_1$ out of $n_1$ people doing $X_1$ died, while only $m_2$ out of $n_2$ people died when doing $X_2$.
I'm trying to understand whether or not the difference between $X_1$ and $X_2$ is indeed significant, so:
Say we have a rare process which occurs with probability $p$. The Poisson limit theorem states that as $n$ becomes large and $p$ becomes small, this process is Poisson distributed with $\lambda=np$.
Now assume we have a single sample data point where $m\ll n$ events occurred.
- Is $p=m/n$ indeed the maximum likelihood estimator?
- What is the probability that $p$ has some different value, i.e., what is: $$P\left(p \mid m,n\right)=?$$
- Given two distributions with different $p_1,p_2$, and a single sample $m_1/n_1$, $m_2/n_2$ for each, is there any way of finding the chances one is "better" than the other, i.e.: $$P\left(p_1>p_2 \mid m_1,n_1,m_2,n_2\right)=?$$
It looks like Bayes's theorem could potentially help, but I don't know what $P(p)$ or $P(m)$ are.
Any ideas on how to decide which option is better?