\cdot \frac{1}{\sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2}} \> . $$ The distribution of a linear combination of $X_1$ and $X_2$ using the same weights as before is, via the stable property of the normal distribution is $$ \varphi_{\ell}(x) = \frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{(x-(1-2\alpha)\mu)^2}{2\sigma^2}} \>, $$ where $\sigma^2 = \alpha^2 + (1-\alpha)^2$. These two distributions are very different,