Suppose you have a series of n trials, where the probability of success in each trial is p. The distribution of the number of successful trials follows a Binomial distribution with parameters (n, p). The mean is given by np whereas the variance is np(1-p). So far so good: this is pretty mundane Stats 101 stuff.
But suppose now that I only knew about the successful trials, and had no knowledge of the total number n of trials, which is the variable I am interested in estimating. For example, I knew I had 100 successful trials, where each trial had a 0.1 chance of success. Is there a known probability distribution that describes the probable outcomes for n, the total number of trials? Estimating the mean is easy: if m is the number of successes, then it's just m/p. But what about variance and other measures?
What if each success had a different (but known) chance of success? Suppose I had the following records:
- success1 (with p=0.1)
- success2 (with p=0.1)
- success3 (with p=0.2)
Again, a good estimation of the total number of trials can be obtained by simply summing 1/p for each successful trial. In this case that number is 10+10+5=25. But what about variance and other measures?