Ask for a question about independence This is the question I met while reading Shannon's channel coding theorem. Assume a random variable $X$ is transmitted through a noisy channel with transition probability $p(y|x)$. At the receiver a random variable $Y$ is obtained. Assume we have an additional random variable $X'$ which is independent of $X$. How to show that $X'$ is independent of bivariate random variable $(X,Y)$ and $X'$ is independent of $Y$? It looks obvious because $Y$ is generated only from $X$, but I just can not prove it rigorously, i.e., that $p(x,y|x')=p(x,y)$ and $p(y|x')=p(y)$. Thanks a lot for your answer!

Two given conditions are:
(1)$X'$ is independent of $X$,
(2)$Y$ is generated only from $X$, i.e., $p(y|x)=p(y|x,x')$. This acutually means $X'\to X\to Y$ forms a Markov chain.
Now let's prove $X'$ is independent of bivariate random variable $(X,Y)$, i.e., $p(x,y|x')=p(x,y)$:
$p(x,y|x')=p(x|x')p(y|x,x')=p(x)p(y|x,x')$(from (1))$=p(x)p(y|x)$(from (2))$=p(x,y)$.
It can easily shown that if $X'$ is independent of $(X,Y)$, then $X'$ is independent of each component. I have shown $X'$ is independent of $(X,Y)$, so $X'$ is also independent of $Y$.
