Generate probability distribution with multiple replications So I have a set of data that represents the times that pets were admitted to the vet. I want to create a probability distribution that models pets being admitted to the vet so that I can use it in a simulation of a day at the vet. 
I have multiple days of admittance times. How do I use the replications to create my probability distribution function?
 A: a) If you assume independent, identically distributed inter-admission times, then you could do one of several things:
i) If you have a large collection of admission times you could simply compute the collection of times between admissions and sample them at random (effectively, sampling them with replacement in the same way as simple bootstrap-sampling is done).
ii) You might assume admissions are independent and uniformly distributed through the day (independence seems like it may be a good approximation to reality, and it might also be the case that uniformity might not be too bad). In that case, you would expect inter-admission times to be approximately exponentially distributed, in which case you could estimate the mean inter-arrival time from the data and use an exponential distribution with that mean in a simulation (you can also check the sensitivity to parameter estimation error by trying other plausible rates and seeing how it impacts the results). I'd do some diagnostics, such a s a Q-Q plot to check the reasonableness of that exponential assumption.
b) if, instead you assume admissions arrive at varying rates (different times of day or different days of the week - etc. - have different admission rates), then you would want a model for your data that tried to estimate the mean rate as a function of whatever factors you expected to have substantive effects (in this case I would be inclined to assume an exponential model for inter-admission times for a couple of reasons). 
These could then be used as a basis for simulation.

Response to information in comments:
As I briefly hinted in my answer, I'd fit a model to the data. 
If you don't expect much variation across days (it doesn't matter what day it is) but only within them (it does matter what time it is), then you can combine data across days.
There are two ways to approach it, you could either fit a Poisson model to the number of admissions ("arrivals") in a small interval of time, or an exponential model for the time between admissions. I'd be inclined to use generalized linear models.
In each case you'd have some kind of trend in the model, but it's not clear to me how this would vary over a day. You may have specific ideas about how that would go.
More generally, if you don't have any specific model in mind for how either the mean admission rate or the expected time between admissions varies over times (only that there are fewer admissions late in the day), you can do something reasonably general:


*

*You might, for example, have a simple model where the rate of admissions per hour (or quarter hour or whatever you like) for a given hour is assumed fixed, each unrelated to the other hours. In fact for a model this simple you don't even need a GLM - you can just tabulate your data for each hour across all days combined and then 
            Total No. of    Number of days 
  Hour:      admissions      in sample
  8.00-         243             39
  9.00-         241             39
 10.00-         221             39
 11.00-         202             39
 ... etc 

to obtain the average admissions per hour across all days. So you might get:
  Hour:    Average admissions per hour:
  8.00-         6.23
  9.00-         6.18
 10.00-         5.67
 11.00-         5.18
 ... etc 

Then when you want to simulate the next hour of admissions, you look up the rate in the table, simulate from a Poisson distribution (what software are you using?) with that rate to get some integer number of admissions (say "8") then uniformly choose the admission times within that hour (by simulating "8" observations uniformly from 0 to 60). [For a first stab at the problem, this is probably about as simple as it gets.]
Alternatively, if you don't have some idea for a model of the way mean admissions change over time, you might fit a smoothly varying model, such as a generalized additive model. This has several benefits but is much less simple, and might be overkill for your purposes.
