Probability distribution of actual time spent if randomly sampled at a known mean rate

Question

I was experimenting with tagtime, which randomly asks the user what they're doing at a known mean rate $\lambda$. Let's say that every time I am sampled, I give a yes/no answer. If I answer yes $k$ times within some period, then I supposedly spent $k\lambda$ in the "yes" state, but clearly this is the mean of the probability distribution of what I was actually doing, since I could have actually spent anywhere from an infinitesimal to an infinite amount of time in "yes." What is the probability distribution of how much time I actually spent in "yes?"

I know a few of the distributions related to this, e.g. that the time between samples follows an exponential distribution with parameter $\lambda$; but not enough to answer my own question.

Answer 1

Requested from comments:

Making the naive assumption that (given the overall proportion of time you are in "yes") what you are doing at any question-time is independent of what you are doing at other question-times, this is essentially estimating a binomial proportion.

If you want a distribution, you may want to use Bayesian methods starting with a Beta conjugate prior for the proportion of time and then rescale to total time. There are many frequentist methods of producing confidence intervals.