please help in providing a proof like explanation. I am confused and do no know where to start.
Let $(X_t)$ and $(Y_t)$ be independent Poisson processes with rates $\lambda$ and $\mu$. Using thinning/superposition, find the probability that the process $(X_t,Y_t)$ ever visits the state $(i,j)$.
This is a straightforward computation:
\begin{align} \int_0^\infty \mathbb P(X_t=i,Y_t=j)\ \mathsf dt &= \int_0^\infty \mathbb P(X_t=i)\mathbb P(Y_t=j)\ \mathsf dt\\ &= \int_0^{\infty } e^{-\lambda t}\frac{(\lambda t)^i}{i!}e^{-\mu t}\frac{(\mu t)^j}{j!}\ \mathsf dt\\ &= \frac{\lambda^i\mu^j}{(\lambda+\mu)^{i+j+1}} \cdot \frac{(i+j+1)!}{i!j!}. \end{align}