Suppose that a certain animal gives birth to a number of offspring $N$. Each child survives with probability $p$ the first year, independently of the other children. Let $Z_{1}$ denote the number of children alive after 1 year. When $N$ is fixed, $Z_{1}$ is binomially distributed, but it is more realistic to take $N$ random.
a) Suppose that $N$ is uniformly distributed on $\{0, 1, 2,\ldots, 10\}$. What is the expected number of offspring that survives after one year?
b) Suppose that $N$ is Poisson distributed with mean $\lambda$. What is the distribution of $Z$ (give a complete proof)?
c) We can generalise the above, by noting that every year the offspring has a survival probability $p$. Only birth is given to the initial $n$ children and no new offspring is introduced. Suppose $N = n$ fixed. What is the distribution of the number of animals alive after 10 years?
d) For $N = n$ fixed, we let $T(n)$ be equal to the first year that all of the offspring is dead. Give the distribution function of $T(n)$.
I managed to figure out b) such that the distribution of Z is $e^{-p\lambda}\frac{(\lambda p)^k}{k!}$. The other questions I cannot figure out. How are these questions supposed to be done?