Note that both diagrams are refferring to the same problem. The difference is that I'm not sure if graph I'm supposed to visualize is the PDF or the CDF, so I drew them both and hope someone will correct me.
In the picture, The Xi are the poisson arrival events. So X1 is the first arrival, X2 the second etc. The parameter is lambda.
The Ti are the interarrival times of successive events. So T1 is the time between the first event and the beginning of time. T2 the time btwn the 2nd event and then first etc. The Ti have an exponential distribution because with rate lambda because we are in a poisson process.
My question is about whether I'm thinking about this correctly. Every time an event happens, is it like we are starting a new poisson process or resetting the exponential distribution? Kind of like, when an event happens, has the process or the exponential distribution renewed? If so, does it make sense to see the PDF or CDF of the Xi as the same PDF/CDF for all Xi? What I mean is, say T1 has an exponential distribution like the one drawn. Then will T2 have the same distribution (since it has the same parameter lambda) except that it will be squished/scaled? (In diagram A I am fitting the PDF of the exponential distribution between each successive event. In diagram B I am fitting the CDF of the exponential distribution between each successive event.)
Also, is diagram A or B correct? Or are they both wrong? Any other problems that you see with my thought process please let me know.
Thank you for your help and my apologies for my bad diagrams in advance.