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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 with independent

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

The kind of mistake this is

Execution slip

The idea was right. The computation was not.

Drill statements like this
statisticsmultivariate

Others that fail the same way

From Linear Models and Multivariate › Multivariate normal distribution

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