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 Xn be the number of individuals that have received the message at time n, en assume that the kth individual with the message at time n, passes it on to Ak,n other individuals. We then have the following recursion for Xn:

Xn = {k=1 to X(n-1) summation A(k,n)} .

Further assume that the random variables Ak,n are independent and identically distributed for all k en n. Use the notation a and v for the (common) mean and variance Ak,n. We assume a > 1 in all questions. 1. Calculate E[Xn|Xn−1] and E[X2 |Xn−1].

2. Calculate E[Xn|X0] en E[X2 n |X0] (tip: calculate first E[X2 n |Xn−2], E[X2 |Xn−3], . . . before you try to get the expression for E[X2 |X0]).