I am taking a Stochastic Processes class and i have a homework assignment that goes like this:
Let X and Y be two homogenous poisson processes with rate 1 and 2 respectively. What is the distribution of $$T=\min(t: X(t)+Y(t)=1)$$
What have i tried so far? I know that $X(t)+Y(t)$ is a Poisson process with rate function $\lambda$ given by $$\lambda(t)=\lambda_{1}(t)+\lambda_{2}(t)$$
Which in our case results in $\lambda = 3$. Moreover, i know that if $Z=\min \left\{X, Y\right\}$, then $P\left(X<Y\right)=\frac{\lambda_{1}}{\lambda_{1}+\lambda_{2}}$, we did that during class. Right now i am trying to do some sort of conditioning to obtain the distribution. How can i proceed?
Any help is appreciated! Thanks in advance