Proof of mutual information property that $I((1-\beta)Z + \beta X; X) \geq I(Z; X)$ Suppose $\beta$ is a Bernoulli random variable taking $\lbrace 0, 1\rbrace$, $X, Z$ are random variables defined on the same probability space. Is it true or false that
$$ I((1-\beta)Z + \beta X; X) \geq I(Z; X) $$

I don't think that this is true. The conditional mutual information 
$$I((1-\beta)Z+\beta X;X|\beta)$$
certainly exceeds $I(Z;X)$, but the same may not hold for mutual information. Consider the following example where $X$ and $\beta$ are independent Bernoulli-$1/2$ random variables taking values in $\{0,1\}$. Let $Z$ take values in $\{0,1\}$ and suppose that $Z=0$ if and only if $X=1$. We hence have $I(Z;X)=1$. However, $(1-\beta)Z+\beta X$ is independent from $X$, hence $I((1-\beta)Z+\beta X;X)=0$.
