I’m studying the Poisson process and have been given a question that I’m having some trouble with. The question is as follows:
Let $\{X(t): t\geqslant 0\}$ be a Poisson process with rate $\lambda = 1$. Find the probability that there are two arrivals in $(0,2]$ and three arrivals in $(1,4]$.
My approach was to define the number of arrivals in three non overlapping intervals: $X_1$ for $(0,1]$, $X_2$ for $(1,2]$ and $X_3$ for $(2,4]$. Then, the number of arrivals in $(0,2]$ is $X_1$ + $X_2$, and in $(1,4]$ is $X_2+X_3$. I tried to calculate the required probability as a sum over the join probabilities for each possible value of $X_2$, taking into account the independent Poisson distributions of $X_1$, $X_2$ and $X_3$.
My approach: current worked solution
However, I’m not completely confident in my solution, and would appreciate any feedback or alternative approaches. Thanks in advance!