A certain organism produces an average of $\displaystyle \mu$ offsprings with variance $\displaystyle \sigma ^2$. Each offspring produces likewise. What would be the expected number of offspring and its variance in the third generation?

If $\displaystyle X_{i}$ = # of offspring in ith generation, I figured $\displaystyle E(X_{3})= \mu ^2$ and $\displaystyle Var(X_{3}) = \mu ^2 \sigma ^2$

I'm pretty sure of expected, but not that sure of variance. I used the fact that if $\displaystyle Y= a+bX$ then $\displaystyle Var(Y)=b^{2}Var(X)$

Can somebody confirm this? Thanks.