2

  1. From here, it says that, linear combination of two Gaussian distribution, are always Gaussians.

  2. However, Let be standard normal and =±1 with probability 1/2 each, independently of . Let =. Then is also standard normal, but =+ is exactly equal to zero with probability 1/2 and is equal to 2 with probability 1/2.

But (2) contradicts with (1). Am I missing anythings?

Nygen Patricia
  • 181
  • 1 is incorrect as a general statement, unless the two have a bivariate/multivariate Gaussian distribution. This point is discussed in the answers and comments on your linked question – Henry Nov 30 '20 at 02:02
  • Henry's comment is true but, also, even if the assumption does hold, you're creating a degenerate case for half of the distribution by adding X to -X so there is no distribution but a point mass at zero. So, it's not really a contradiction, atleast to me. When they talk about linear combinations of normals, they are referring to linear combinations of DIFFERENT normals rather than the same one. – mlofton Nov 30 '20 at 05:02

0 Answers0