Is it true:$C = C_1 * C_2$ where $C_1$ and $C_2$ are the two cascaded BSC which makes $C$.
No, this is not correct. Consider the chain of two BSCs with error probabilities $p_1$, $p_2$ as a single BSC with unknown error probability $p$.
Now, we know that in overall no error occurs, when:
- neither BSC1 nor BSC2 create an error: $(1-p_1)(1-p_2)$
- both channels make an error: $p_1p_2$.
Hence the overall probability of an error is
$$ p= 1-[(1-p_1)(1-p_2)+p_1p_2]=p_1+p_2-2p_1p_2 $$
which is also known as the binary convolution. Now, the capacity of the chain-BSC is given by $$C=1-H(p)$$ where $H(p)$ is the binary entropy function.