How to prove that a mixture of Gaussians is Gaussian?

Question

Consider independent random variables $\mathbb{P}(X=-1) = \mathbb{P}(X=1) = 0.5$ and $Y \sim \text{N}(0,1)$ and let $c \neq 0$ be a real constant. How do I prove that $cXY$ is Gaussian ? What is the expectation and variance?

Answer 1

$Z=XY$ is $cY$ with $0.5$ probability and $-cY$ with $0.5$ probability. $cY=T$ is a Gaussian, so as $-T$, because a scalar non-zero multiplication with a Gaussian produces a Gaussian. Also, because $T$'s PDF is symmetric around $0$, they have the same PDF, i.e. $f_T(t)=f_{-T}(t)$. Using mixture properties, you'll have $$f_Z(z)=0.5\times f_T(z)+0.5\times f_{-T}(z)=f_T(z)$$

$f_T(z)$ is Gaussian PDF with mean zero and variance $c^2$, so is $f_Z(z)$.