Is the following statement true? If every marginal is normal then the vector is multivariate normal

Counterexample

Is this true?

If every marginal is normal then the vector is multivariate normal

No — it is false.

The counterexample

X ~ N(0,1) and $Y = \varepsilon{}X$ with $\varepsilon = \pm1$ independent

Why it fails

Both marginals are N(0,1) but X + Y is 0 half the time — not normal, so the pair is not jointly normal.

Keep practicing

Boundary and endpoint

The statement turns at the edge of the interval, the domain, or the parameter range.

Published 23 Aug 2026. Found a mistake? Tell us — corrections are the fastest thing we act on.