Is the following statement true? If X and Y are each normal and uncorrelated then they are independent

No. X ~ N(0,1), ε = ±1 with probability ½ independent of X, Y = εX Y is N(0,1), Cov(X,Y) = 0, but |X| = |Y| always. The pair is not jointly normal.

Counterexample

Is this true?

If X and Y are each normal and uncorrelated then they are independent

No — it is false.

The counterexample

X ~ $N(0,1), \varepsilon = \pm1$ with probability ½ independent of $X, Y = \varepsilon{}X$

Why it fails

Y is N(0,1), Cov(X,Y) = 0, but |X| = |Y| always. The pair is not jointly normal.

Keep practicing

Standard counterexample

There is a canonical object that settles this. Recognising it is the whole question.

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