Poisson Process Arrival Time

Question

In a poisson Process with λ = 5 .. and Ti represent the time of ith arrival .. Find ℙ(T3 < 1.1 | T1 > 0.5).

I don't know whether they are independent or not.

Answer 1

You need the Erlang distribution. Independence of the underlying events is the foundation of the Poisson process model but the cumulative time for a number of events is not determined directly from an independence argument.

If you mean the independence of the two clauses, no they are not independent. If T1 is large, T3 is more likely to be large. Consider the same process on a different time scale, 1/10 second bins, lambda = 0.5. Bin counts are still independent. Zeros in the first five bins implies a reduces likelihood that there will be 3 or more events accumulated in the first 11 bins.

Answer 2

This is the probability $p$ that there are at least three arrivals in the time interval $(0,b)=(0,1.1)$ conditionally on the event that there is none in the time interval $(0,a)=(0,0.5)$. The Poisson process has independent increments hence $p$ is the probability that there are at least three arrivals in the time interval $(a,b)$. The number of arrivals for a Poisson process of intensity $\lambda$ in an interval of length $b-a$ is Poisson with parameter $c=\lambda\cdot(b-a)$ hence $$ p=\sum_{n\geqslant3}\mathrm e^{-c}\frac{c^n}{n!}=1-\mathrm e^{-c}\cdot(1+c+\tfrac12c^2), $$ with $c=5\cdot(1.1-0.5)=3$.