Let $X_n, n \in \mathbb Z_+$, be the branching process generated by $\xi$, $\mathbb E \xi = a$ $\mathbb Var \xi = \sigma^2$. Find the $E X_{n} $ and $Var X_n$
My Attempt: By the Galton-Walton process, $$X_n = \sum_{i=1}^{X_n -1} \xi_{i}^{n}$$ The Expectation is defined as: $$ EX_n = E(E \left( X_n|X_{n-1} \right) )$$ $$E \left( X_n|X_{n-1} =k \right) = E \left( \sum_{i=1}^{X_n -1} \xi_{i}^{n}|X_{n-1} =k\right)=ka $$ $$\implies EX_n = E(E \left( X_n|X_{n-1} \right) ) = a EX_{n-1}$$ Since, $EX_1 =E\xi =a \implies EX_n =a^n$
Please I am stuck here. Is my approach correct? Also any help or hint on how to find the Variances is appreciated. Thanks!