Joint Normality

Uncorrelated jointly normal implies independence.

But two uncorrelated normal does not imply jointly normal, nor independence.

Consider X ~ N(0, 1) $\( Y = \begin{cases} X & \frac{1}{2}, \\ -X & \frac{1}{2}, \end{cases} \)$

Y is normal, and in fact Y ~ N(0, 1)

\[ \text{Var}(Y) = \mathbb{E}[Y^2] - (\mathbb{E}[Y])^2, \]

\[ \mathbb{E}[Y^2] = \frac{1}{2} \mathbb{E}[X^2] + \frac{1}{2} \mathbb{E}[(-X)^2] = 1, \]

\[ \text{Var}(Y) = 1 - 0^2 = 1. \]

But X and Y are not independent, and (X, Y) is not jointly normal

To be jointly normal, any linear combination aX + bY must be normal

Note P(X + Y = 0) = 1/2, so clearly X+Y is not normal