Suppose I have two Gaussian random variables $Z_1$ and $Z_2$ with mean zero but different variances. I have a mixture \begin{align} Y=Z_1B+Z_2(1-B), \end{align} where $B\sim\text{Bernoulli}(p)$. In other words, I have $Y=Z_1$ with probability $p$ and $Y=Z_2$ with probability $1-p$. Is $Y$ still a Gaussian? If so, how do I show it?
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2It is not. It's a gaussian mixture model(https://en.m.wikipedia.org/wiki/Mixture_model) – J. Delaney Feb 17 '22 at 18:29