What is the distribution of $\frac{X_1}{X_2}$ if both $X_1$ and $X_2$ follows the Poisson Process?

Question

What is the distribution of $\frac{X_1}{X_2}$ if both $X_1$ and $X_2$ are the Poisson Processes with parameters $\lambda_{1}$ and $\lambda_{2}$ respectively? Please list the properties used too. Thanks !

Answer 1

$$P\left(\frac{N_1(t)}{N_2(t)}\leq x|N_2(t) \geq 1\right)=P\left(N_1(t)\leq N_2(t)x |N_2(t) \geq 1\right)=$$ $$\sum_{i=1}^\infty P\left(N_1(t)\leq ix\right)P(N_2(t)=i |N_2(t)\geq 1)=\sum_{i=1}^\infty \sum_{j=0}^{[ix]} P\left(N_1(t)=j\right)P(N_2(t)=i|N_2(t)\geq 1)$$ $$=\sum_{i=1}^\infty\sum_{j=0}^{[ix]} P\left(N_1(t)=j\right)\frac{P(N_2(t)=i)}{P(N_2(t)\geq 1)}$$ $$=\frac{1}{1-e^{-\lambda_2t}}\sum_{i=1}^\infty\sum_{j=0}^{[ix]} P\left(N_1(t)=j\right)P(N_2(t)=i)$$