title: Are Gaussian clusters linearly separable?
body: <p>Imagine you have two Gaussian probability distributions in two-dimensions The first is centered at (0,1) and the second at (0,-1). (For simplicity, assume they have the same variance.) Can one consider that the clusters of data points sampled from these two Gaussians are linearly separable?</p>
  
  <p>Intuitively, it's clear that the boundary separating the two distributions is linear, namely the abscissa in our case. However, the formal requirement for linear separability is that the convex hulls of the clusters do not overlap. This cannot be the case with Gaussian-generated clusters since their underlying probability distributions pervade all of R^2 (albeit with negligible probabilities far away from the mean).</p>
  
  <p>So, are Gaussian-generated clusters linearly separable? How can one reconcile the requirement of convex hulls with the fact that a straight line is the only conceivable "boundary"? Or, perhaps, the boundary effectively ceases to be linear once non-equal variances come in the pictures?</p>