From here, it says that, linear combination of two Gaussian distribution, are always Gaussians.
However, Let 𝑋 be standard normal and 𝜀=±1 with probability 1/2 each, independently of 𝑋. Let 𝑌=𝜀𝑋. Then 𝑌 is also standard normal, but 𝑍=𝑋+𝑌 is exactly equal to zero with probability 1/2 and is equal to 2𝑋 with probability 1/2.
But (2) contradicts with (1). Am I missing anythings?