We have a positively valued known random variable $X$ from 0 to infinity. And a predetermined large number $T$.
What is the probability distribution (or approximation, or key values including mean and variance):
of the number of independently valued 'steps' in the series (the sum, S) of randomly selected values from X required to 'Reach' $T$. Since $T$ is large, it is going to be a well defined probability distribution depending on $T$. In other words, we need the probability distribution of the smallest $N$ such that $S_N \geq T$
One key point is that the expected value of this new distribution cannot be just the sum divided by the mean of the original distribution, it has to be less than than. For example, in Brownian motion we can select a step value -1,1 uniformly distributed and iterate and set a sum value of 5 and get all the step sequences and take the average number of steps N computationally. However using division by mean technique, the expected value would be infinity. **
This problem has many applications. For example, if a phone company A bills users by second, and phone company B bills users by minute starting at 0. If we are given the distribution of phone call lengths. assuming that across all phone calls, the distance to the ceiling of the time by minute is a uniform distribution from 0 to 60 seconds, then each phone call (we can further split out the proportion of phone calls that last only 10 seconds which might skew that mean forwards) adds an expected 30s of time for company B. Then we need to know how many phone calls on average you have for large amount of call time, and then multiply this number by 30s to find the extra time billed by phone company B. and then use the adjusted times to adjust their relevant rates.
Further knowledge about the distribution can give uncertainties and approximations of these values.
** doesn't quite apply because negative numbers are involved