In a Markov process, a particles give birth at the points of a poisson process with intensity $\lambda$ and die at rate $\kappa/N$ and become infertile at per particle rate $\gamma$. What is the probability that the particle gives birth at least once before it dies or becomes infertile?
My attempt is as follows:
Let the fertility periods be $L\sim exp(\gamma)$ and life expectancies be $K\sim exp(\kappa)$. The time period until death or infertility is $Z=min\{L,K\}$. Now $Z>z\iff L>z\land K>z$ so we have that $z$ has cdf $F_Z(z)=1-e^{-(\gamma+\kappa)}$ and pdf $(\gamma+\kappa)e^{-(\gamma+\kappa)z}$. The probability of having at least 1 child is the 1-probability of no children so the probability is $1-\int_0^\infty e^{-\lambda t}\mathbb{P}(Z\leq t)dt=1-\int_0^\infty e^{-\lambda t}(1-e^{-(\gamma+\kappa)t})dt=1-\frac{\gamma+\kappa}{\lambda(\lambda+\gamma+\kappa)}$
Is this correct?