Distribution of the linear combination of two perfectly correlated gaussian

Question

Let us assume that $X$ is normally distributed $$X\sim N(\mu,\sigma^2)$$ If the R.V. $W$ has distribution $$F(w)=P(X \leq w)0.5+P(-X \leq w)0.5.$$ Is $W$ normal? I don't think so, but how do I show it? And what if $$F(w)=P(X \leq w) 0.5 \text{ ?}$$

Answer 1

I don't think that $W$ is normal in general. The below is the beginnings of a solution. First note that we can construct the distribution of $W$ as follows. Let $Z$ be a bernoulli random variable with $p=1/2$, that is independent of $X$. Then W is equal in distribution to $XI(Z=1)+(-X)I(Z=0)$ where $I$ is the indicator function. Put in this fashion we can compute the characteristic function of $W$ to be $$ \phi_{W}(t)=Ee^{itW}=\frac{1}{2}\phi_{X}(t)+\frac{1}{2}\phi_{X}(-t) $$ where $\phi_{X}(t)=\exp(i\mu t-\frac{1}{2}\sigma^2t^2)$.

If $\mu=0$, then $\phi_{W}=\phi_X$ so $W$ is equal in distribution to $X$. If $\mu\neq 0$, then $$ \phi_{W}(t)=\frac{1}{2}\exp\left(\frac{1}{2}\sigma^2t^2\right)(e^{\mu t}-e^{-\mu t}) $$ and it doesn't seem like you can put it in a form to resemble a normal distribution characteristic function in general.