Are the number of events in a given interval always independent of one another in a Poisson Process? I understand that waiting-time is memoryless, so I can just imagine calculating the expected number of arrivals if the interval starting time was $T_0$. But what if the starting time was within another interval?
For example, if I am told that the rate is 5 events/period and that 100 events occurred between $T_0$ and $T_{10}$, and I need to calculate the expected number of events between $T_5$ and $T_{10}$, how would I go about doing this? Is $E[T_{10} - T_5]$ independent of the 100 events or no? Because if it was, then I assume the answer would be 25, but this doesn't seem correct given the context.
 A: welcome to SE. First of all, I infer that with expression "T0" you meant "T=0", and so on.
On your question in your first paragraph: The memorylessness property of the Poisson process means that that the answer is "yes" – the starting time need not be a point of the process (a point where an event occurred). 
On your question in your second paragraph: the hunch that you express in your final sentence is right. The crucial difference to the situation in the first paragraph is that you already know the number of arrivals in an interval – in your case the interval I = [0,10] – containing the interval about which you want to say something, that is the interval J = [5,10] in your case. You are interested in a conditional expectation given the observation of 100 points in I. You will see in a moment that the initial information on the rate is irrelevant in computing this conditional expectation. 
Each of the 100 observed points, which we think of here as unordered, has a probability of (length of J)/(length of I) = 0.5 of falling in the interval J, independent of the other points. Thus the conditional expectation of the number of points falling in J is 50. This is much higher than the unconditional value of 25, which is because the event on which you conditioned, namely that 100 points fall in J (rather than just 50 or even, say, 55), has such a small probability. 
