What is the degenerated normale law?

Question

In my lecture, my teacher always talk about non-degenerate normale law. But what would be a degenerated normale law ?

Answer 1

Any constant random variable $X=c$ is considered as a normal random variable with mean $c$ and variance $0$. These are called the degenerate normal random variables.

Note that when are dealing with linear combinations of jointly normal variables you are always into the problem of getting a constant (like (1)X+(-1)X=0$. For that reason it is helpful to define constants as normal random variables.

Non-degenerate normal means genuinely normal, not a constant.

In the case of Guassian $n-$ vectors non-degenerate means that the values of the random vector do not line in any proper affine subspace. This is so iff the variance - covariance matrix is non-singular.