Best distribution to model leaving time of people at a concert I'm writing a simple model, modelling the leaving time of people at a concert. Data shows that people are most likely to leave just after the concert finished, but some leave earlier and some stay longer at the venue. I'm wondering which distribution best describes the leaving time, measured since the starting time of the concert. The length of the concert is given (but people may choose to stick around after).
Things I've thought of:

*

*Normal(mu, sigma) with unknown mu and sigma. This seems not ideal since we know that people don't leave before the concert has started, ie time of leaving will never be negative.

*Lognormal or Gamma, with unknown parameters. Their pdf seems to have the 'right' shape, 0 for negative values with a bump around a specific time. I know both are max-entropy for specific statistics given. Can I use this for making the correct choice?

How do I pick the best distribution to model this? What principles do I use to choose?
Add-on to the question: in a bayesian setting, which would be appropriate priors to use for the parameters? I know that for the mean and var of a Normal, we often use a Normal and a inverse Gamma prior respectively, but I have little intuition as for why (esp in the case of the inverse gamma).
If there are no direct answers to this question, eg both lognormal or gamma would be good for different reasons, that is also a great answer to know.
 A: There is no general answer to this question. Every model
is wrong but some are useful.
You have to fit different distributions. Then compare how good you can represent the data (compare real histogram vs predicted histogram and compare cumulative distributions). The analytic distribution can be evaluated at the $x$ values from your empirical data. You can then calculate the mean squared error, mean absolute error, etc.
In this procedure you should also use cross validation to reduce risk of overfitting.
I would first try a standard parametric model and then consider more complicated models if the performance is not sufficient. At first glance using Bayesian Optimization is an overkill since you still have to come up with a likelihood and prior.
A: Propose a model. One possibility out of many is $X - 1,$ where $X \sim\mathsf{Gamma}(2, 1).$ My purpose is to demonstrate an approach toward modeling
this situation--not to give a definitive answer.
According to my proposed model,
some leave up to a minute early, most leave just as the
concert finishes and most are gone 9 or 10 minutes after
the concert is over. Parameters would depend on the size
of the audience, the accessibility of exits, and the
formality of the event.
For details on gamma distributions, see Wikipedia.
Plot its density function. A graph of the density function of $Y = X - 1$ [with support
$(-1,\infty)]$ follows.
curve(dgamma(x+1,2,1), -2, 10, lwd=2, 
      ylab="Densitty", xlab="x-1", main="Leaving Times")
 abline(h=0, col="green2")


Simulate to help judge whether the model is realistic. Simulation of leaving times y for an audience of 1000, perhaps at an outdoor concert, where people
try to avoid the inevitable traffic jam getting out of the parking area. You may have a different kind of concert in mind.
set.seed(2021)
x = rgamma(1000, 2, 1)
y = x-1
sum(y < 0)
[1] 240      # number leaving just before end
sum(y > 5)
[1] 25       # number leaving more than 5 min after

summary(y)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
-0.9462  0.0218  0.7447  1.1234  1.8564  8.1892 

