the definition of normal vector

Question

I got confused with definitions of a normal vector.

Assume, that $X \in \mathcal N(\mu, \sigma^2)$ and let $Y = 1 - X$. Is this true that $(X, Y)$ is multivariate normal?

The problem is that according to the first definition of a multivariate normal vector, any linear combination of the components must be a normal random variable. In the example above we have that $P[X + Y = 1] = 1$. Is this a contradiction?

Answer 1

There is no contradiction here.

Due to the way the RVs are defined, the joint variable have a singular covariance matrix. This is the reason why the sum is equal to a constant.

In this case, the joint variable does not have a density; see https://en.wikipedia.org/wiki/Multivariate_normal_distribution#Definition https://en.wikipedia.org/wiki/Multivariate_normal_distribution#Degenerate_case