We study the diffusion of a message (say, like a tweet) on a social network. To this end, we use the following simplified model. Let $X_n$ be the number of individuals that have received the message at time $n$, then assume that the $k^{th}$ individual with the message at time $n$, passes it on to $A_{k,n}$ other individuals. We then have the following recursion for $X_n$:
$$X_n=\sum_{k=1}^{X_{n-1}}A_{k,n}$$
Further assume that the random variables $A_{k,n}$ are independent and identically distributed for all $k$ and $n$. Use the notation $a$ and $v$ for the (common) mean and variance $A_{k,n}$. We assume $a > 1$ in all questions.
Calculate $E[X_n|X_{n−1}]$ and $E[X_2|X_{n−1}]$.
Calculate $E[X_n|X_0]$ and $E[X_2|X_0]$.
(tip: calculate first $E[X_2|X_{n−2}]$, $E[X_2|X_{n−3}]\dots$ before you try to get the expression for $E[X_2|X_0]$)
Can anyone help me how to solve it when i dont have value for $X_0$ and $k$ starts from $1$?