Exercise $1.$ Separating two Gaussians

Suppose you have two Gaussians located at $$ and $-$ respectively, i$.$e$.,$ Gaussian$(x$;$,{}^2)$ and Gaussian$(x$;$-,{}^2).$

We know that if $$ is large enough, then the two Gaussians will be separated. We also know that if $$ is

small enough, the two Gaussians will be separated. Fix $.$ What is the largest $$ you can have before the

two Gaussian become merged together? By merge, I mean that you will no longer be able to see the two

peaks. The figure below is an example for $=0\mathrm{.}5.$ As you can see, when $=0\mathrm{.}5,$ the two Gaussians cannot

be distinguished. Use paper and pencil to answer this question, and use Python$/$MATLAB to verify your

answer.