Say we have a population of people. Each person in the population performs an action every day, with probability X. In other words, on a given day, there is an X percent chance that specific person performs the given action.
Suppose that we know the distribution of X across the population. Ie, we know that 5% of people have an X of 10%, 7% of people have an X of 20%, etc. This distribution is not a normal distribution.
Now, the actual question... Let’s say we have one specific person, and I have a very small bit of data about that person. Let’s say I only have 3 days of data about him. On all 3 of those days, that person performed the given action. How can I use the known population distribution, and the tiny sample size of information about the specific person, to estimate the value of X for that specific person?