Question: Given $X\sim\text{Bernoulli}(\alpha)$, $Y\sim\mathcal{N}(0,1)$, and non-random positive constants $C,\epsilon>0$. Let $H(\cdot)$ be the differential entropy. Is it true that $$ H((C+\epsilon)X+Y)>H(CX+Y), $$ and is there a rigorous way to show this?
Attempt: Intuitively I suspect this to be true since from here, we know that scaling a random variable $W:=CX+Y$ introduces an additive factor. The issue here is that we are technically only (up)scaling a part of $W$ instead of the entire $W$.
If we are to think of $W$ as a mixture of two Gaussians, then the $\epsilon X$ additive component is akin to shifting on of the Gaussians.
