Question:
In a given population, each individual has a number of offspring $Y$ with discrete Uniform distribution on $\{0, 1, \ldots, N \}$, for some fixed $N \ge 1$, i.e. $P(Y = k) =\frac{1}{N+1}$ for all $k = 0, 1, \ldots, N$. Let $\pi_N$ denote the probability of eventual extinction of this population when it starts from one individual.
Find a formula involving $\pi_N$ for the extinction probability of this population if its initial number of individuals $X_0$ is random and follows the P(λ)-distribution (Poisson).
Any help would be appreciated!
The population goes extinct with probability $\pi_N^{x_0}$. Thus its extinction probability is
$$ \sum_{x_0=0}^\infty P(X_0=x_0)\pi_N^{x_0}=\mathrm e^{-\lambda}\sum_{x_0=0}^\infty\frac{\lambda^{x_0}}{x_0!}\pi_N^{x_0}=\mathrm e^{-\lambda}\mathrm e^{\lambda\pi_N}=\mathrm e^{-\lambda(1-\pi_N)}\;. $$