Me and my teammate have this question to solve in our homework. The question is the following:
A N(t) non homogeneous Poisson process with intensity $\lambda$(t) = $\frac{1}{1 + t}$. Find the density of $T_1$ as well as the density of $T_2 - T_1$.
I have seen various notations, so in order to avoid any confusion I'll the definitions of the notation I have seen in class:
m(t) = $\int_0^{t}\lambda(u) du$
Finding the density of $T_1$ was fairly easy. All me and my teammate had to do was find
1 - P[$T_1$ > t] = 1 - P[N(t) = 0] = 1 - $e^{-m(t)}$ = F$_{T_1}$
f$_{T_1}$ = $\frac{d}{dt}$F$_{T_1}$ = $\lambda(t)e^{-m(t)}$
Our issue is with f$_{T_2 - T_1}$.
We found this post from 11 years ago that has a similar problem (Arrival times, joint density) (the original op want to find f$_{T_1,T_2 - T_1}$ instead of just f$_{T_2 - T_1}$. One of the answers of the post says that in order to find f$_{T_1,T_2 - T_1}$ we can evaluate it by doing:
f$_{T_1,T_2 - T_1}$ = f$_{T_1}$*f$_{T_2 - T_1|T_1}$.
Finding f$_{T_2 - T_1|T_1}$ is also relatively easy. Here's our solution:
P[$T_2 - T_1 > s| T_1 = t]$ = P[N(t + s) - N(t) = 0]
P[N(t + s) - N(t) = 0] = 1 - $e^{-m(t+s) + m(t)}$ (N(t + s) - N(t) follows a Poisson distribution with parameter m(t+s) - m(t))
f$_{T_2 - T_1 | T_1}$ = $\lambda(t + s)e^{-m(t+s) + m(t)}$
All we have left to do is find f$_{T_1,T_2 - T_1}$. By using the formula mentionned earlier (of which I don't understand how we can do that) we find:
f$_{T_1,T_2 - T_1}$(t,s) = $\lambda(s + t)\lambda(t)e^{-m(t + s)}$
Since we have the joint density all we have left to do is integrate this function by t and we'll get f$_{T_2 - T_1}$.
f$_{T_2 - T_1}$ = $\int_0^{\infty}\frac{1}{(1 + t + s)^2(1 + t)}dt$ (m(t) = ln(1 + t))
Using this website to solve this integral (https://www.integral-calculator.com/), we find the following density:
f$_{T_2 - T_1}$ = $\frac{(s + 1)ln(s + 1) - s}{s^2 + s^3}$
Now me and my teammate are wondering if what we did here makes ANY sense at all (and if it does is our answer even good?) and if there's an easier way to solve this question without using the joint and conditionnal density.