Consider three independent Poisson processes $N_1(t), N_2(t),N_3(t)$ with rates $\lambda_1,\lambda_2,\lambda_3$, respectively. What is the probability of reaching $(i,j,k)$?
I know that for the two-dimensional process, we have the probability of reaching $(i,j)$ as $$\frac{(i+j)!\lambda_1^i\lambda_2^j}{i!j!(\lambda_1+\lambda_2)^{i+j}}$$ Could the problem be easily extended to the three-dimensional Poisson process as $$\frac{(i+j+k)!\lambda_1^i\lambda_2^j\lambda_3^k}{i!j!k!(\lambda_1+\lambda_2+\lambda_3)^{i+j+k}}$$
I have reason to believe so since by considering $\lambda_1=\lambda_2=\lambda_3=1$; $i=j=k=n$, we have the probability of reaching $(n,n,n)$ as $$\frac{3n!\left(\frac{1}{3}\right)^{3n}}{n!n!n!}$$