I have a quick question on the following exercise:
Let $C = (X , P, Y)$ be a binary channel which is composed of two binary channels in sequence, such that the output of the first channel $C1 = (X , P_1, Z)$ is the input of the second channel $C_2 = (Z, P_2, Y)$. Let the transition matrices be given by
$$P_1 := \begin{bmatrix} 3/4 &1/4\\ 3/4 &3/4 \end{bmatrix} \quad \text{ and } \quad P_2 := \begin{bmatrix}1/3 &2/3\\ 2/3 &1/3 \end{bmatrix}$$
Compute the transition matrix $P$.
I just want to make sure that I am on the right track: $P$ should be given by $P = P_2 \cdot P_1$, since $\mathcal{C_1}$ maps into $\mathcal{C_2}$, right?
Think in terms of the variables. $C_1$ maps $X \to Z$ with respect to $P_1$ and $C_2$ maps $Z \to Y$ with respect to $P_2$.
Since $C$ maps $X \to Y$, the equivalent cascaded channel is $X \to Z \to Y$. First, $X$ passes through $C_1$ (and hence $P_1$) and then the obtained $Z$ passes through $C_2$ (and hence $P_2$). Following the order, we get $P = P_1 P_2$, assuming $C_1$ and $C_2$ are independent of each other.