I am looking at waiting times between two events from multiple patients, so I'm looking at a gamma distribution. Turns out, the model is plotting out an exponential distribution, which if I was to look at something like the arrival times of ambulances at A&E this would make sense. However, I'm now unsure whether I am modelling my data correctly, as I am only looking at a single time-to-event per person, each person being mutually exclusive from the next, and exponential distributions are meant to be memoryless. Surely, any data is memoryless if there is only one event per independent and mutually exclusive case?
The scenario: I would like to plot the probability distribution of the time from a headache diagnosis (H) made in a Doctors (GP in the UK) to a headache diagnosis made at a special referral clinic (R). This is on a per patient basis e.g.,
Is it correct to try modelling this kind of data using a gamma distribution and then would it be logical to end up with an exponential distribution?