Unique Properties of Gaussian RVs

Last modified: July 21, 2026

  1. All linear combinations of independent Gaussian RVs are Gaussian

  2. A Gaussian random vector is a random vector all of whose linear projections are Gaussian.

  3. If $\underline{X}$ is Gaussian , then

    1. $X_{i}$ and $X_{j}$ are independent $equivalent$ Cov(Xi,Xj)=0 (uncorrelated)

  4. RVs are jointly Gaussian if they constitute a Gaussian random vector.

  5. It is possible for random variables to be individually Gaussian but not jointly Gaussian. Jointly Gaussian is a stronger property.