## Statistics and Markov Chains

Why does $\text{Var}(E[N_i|S_i]) = \text{Var}(E[N_i|S_i]-1) = (\mu_{i-1}+\mu_i)^{2}q - (\mu_{i-1} + \mu_{i})^{2}q^2$?

The point is, we are trying to compute $\text{Var}(N_{0,n})$ or the variance of the expected number of transitions needed to go from state $0$ to state $n$.

This is from the book by Ross (An Introduction to Probability Models).