I need to compute the following joint distribution in a Poisson process:
$f_{S_A S_{A+B}}(t_1, t_2), t_2\ge t_1$
$S_A$ and $S_{A+B}$ are the arrival epochs of the $A^{th}$ and ${A+B}^{th}$ arrivals respectively. I tried the following:
\begin{align}
P(S_A\ge t_1, S_{A+B}\ge t_2)&=P(N_{t_1}\le A, N_{t_2}\le A+B)\\
&=P((N_{t_1}\le A)P(N_{t_2-t_1}\le B)\\
&=\sum_{k=0}^A \frac{e^{-\lambda t_1} (\lambda t_1)^k}{k!}\sum_{l=0}^B \frac{e^{-\lambda (t_2-t_1)} (\lambda (t_2-t_1))^l}{l!}
\end{align}
To find the density, I will need to differentiate this wrt $t_1$ and $t_2$. However the expressions are unwielfy. Is the approach correct? Is there a simpler way to arrive at the solution?
Best Answer
Hint: The random variables $S_A$ and $S_{A+B}-S_A$ are independent and $S_{A+B}-S_A$ is distributed like $S_B$. Now, you ought to know the distribution of each $S_A$ hence you ought to be able to finish this.