# Poisson arrival times joint distribution

The arrival times of the first and second event are $S_1$ and $S_2$, and the number of arrivals follow a poisson process. How would I compute the Joint PDF of $S_1$ and $S_2$?

I have found the PDF of $S_1$ and $S_2$: $f_1(s)=\lambda e^{-\lambda s}$ and $f_2(s)=\lambda^2 se^{-\lambda s}$.

I also know the variables $S_1$ and $S_2-S_1$ are independent.

• Sorry, I differentiated the CDF incorrectly, but I will correct the original post. – Timothy Hedgeworth Apr 23 '15 at 15:27

I also know the variables $S_1$ and $S_2-S_1$ are independent.
Yes, and thus: \begin{align} f_{S_1,S_2}(s_1, s_2) & = f_{S_1, S_2-S_1}(s_1, s_2-s_1) \\[1ex] & = f_{S_1}(s_1)\cdot f_{S_2-S_1}(s_2-s_1) \end{align}
Now, do you know what $\;f_{S_2-S_1}(t)\;$ is?
• I am pretty sure: $f_{S_2-S_1}(t) = \lambda e^{-\lambda t}$. So would $f_{S_1,S_2}(s_1, s_2)=f_{S_1, S_2-S_1}(s_1, s_2-s_1)=\lambda^2 e^{-\lambda s_1} e^{-\lambda (s_2-s_1)}=\lambda^2 e^{-\lambda s_2}$? – Timothy Hedgeworth Apr 23 '15 at 15:51
• @TimothyHedgeworth Yes, indeed -- for $0 < s_1 < s_2$ ; it's $0$ elsewhere. – Graham Kemp Apr 23 '15 at 16:18