# Poisson distribution probability from a single measurement

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?

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Indeed without a prior you are lost. Not sure why Poisson distributions are mentioned at all but 1. the MLE for the parameter of a binomial distribution is trivial, just differentiate the PMF, and 2. the usual conjugate priors of binomial distributions are beta distributions, see en.wikipedia.org/wiki/Conjugate_prior –  Did Feb 8 at 8:23
@Did - there must be some way. Clearly when $n\to\infty$ if $m_1\ll m_2$ there is a lot to be said for treatment $X_1$. The question is, just how much. –  nbubis Feb 8 at 8:34
Yeah, and to answer "how much", one must rely on something, most notably a prior. –  Did Feb 8 at 10:18
@Did could you expound on the topic? What would be a reasonable choice for a prior, and how would one use it? What I knew about about priors and bootstraps vanished a long long time ago. –  nbubis Feb 8 at 19:27
The usual conjugate priors of binomial distributions are mentioned in my first comment and explained in the link in this comment. –  Did Feb 8 at 22:32