Mean and variance of sum of random variables with known contingent distributions

Question

Let $X_0, X_2,...,X_n$ be integer valued random variables ($0$ also allowed) with the property that $X_0 \geq X_1 \geq ... \geq X_n$, let $a_0, a_1,..., a_n$ be real numbers and $p_0, p_2,..., p_{n-1} \in (0,1)$ . Assume that $X_0 = x_0$ almost surely for some fixed number $x_0 \neq 0$ and that $X_j \mid X_{j-1}=x_{j-1} \sim binomial(x_{j-1}, p_{j-1}) $ for $j=1,2,...,n$ ( by "$\mid$" I mean contingent distribution). My question is, can anyone tell me how I calculate mean and variance of the sum \begin{align} \frac{1}{X_0}\sum_{j=0}^n a_j X_j \end{align}

Answer 1

In general your expression has mean $a_0+p_1(a_1+p_2(a_2+\cdots p_{n-1}(a_{n-1}+p_n a_n)\cdots)))$ from the obvious pattern. The variance will not be so simple