Notes on the MVN

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What is the MVN!?

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A random vector $\mathbf{X} = (X_1, ... , X_k)$ is MVN if every linear combination is a Normal RV.

Example that is MVN: Let $Z$ and $W$ be iid $N(0,1)$ then the vector $\mathbf{X} = (Z,W)$ is MVN. Also, $\mathbf{X} = (Z + 2W, 3Z +5W)$ is also a MVN. You can see this because $s(Z + 2W) + t(3Z + 5W) = (s+3t)Z + (2s+5t)W$ which is the sum of two independent Normals which is a Normal RV.

Example that is NOT MVN: Let $Z \sim N(0,1)$ and let $S$ be a random sign independent of $Z$. Then marginally $Z$ and $SZ$ are both standard Normal. However the pair $\mathbf{X} = (Z,SZ)$ is not MVN. For example look at the linear combination $Z+SZ$ and note that half the time you get $0$ and the other half you get something continuous. This linear combination is clearly not a Normal RV. This example shows that a MVN is not just about stringing together any Normals RVs.

MGF of $\mathbf{X} \sim MVN$

$$M_{\mathbf{X}} (\mathbf{t}) = E[e^{\mathbf{t} ' \mathbf{X}}] = E[e^{t_1X_1 + ... + t_k X_k}] = exp \left\{ t_1 \mu_1 + t_k \mu_k + \frac{1}{2} Var(t_1X_1 + ... + t_k X_k) \right\}$$

where we used the fact that the exponent is a Normal RV since $\mathbf{X}$ is MVN.

Within a MVN, uncorrelated implies independence!

Let's say you have $\mathbf{X} = (\mathbf{X}_1, \mathbf{X}_2)$, if every component of $\mathbf{X}_1$ is uncorrelated with every component of $\mathbf{X}_2$, then they are independent. So take any component of $\mathbf{X}_1$ and look at the covariance with any component of $\mathbf{X}_2$. If they're all zero then $\mathbf{X}_1$ and $\mathbf{X}_2$ are independent.

Example that doesn't work: Note that $Z$ and $SZ$ are uncorrelated but not independent. They are dependent but have covariance of zero. Of course $(Z, SZ)$ is not MVN.

Example that does work: Let $X$ and $Y$ be iid $N(0,1)$ and let $\mathbf{X} = (X+Y,X-Y)$. First note that $\mathbf{X}$ is MVN. It turns out $X+Y$ and $X-Y$ are uncorrelated since $Cov(X+Y,X-Y) = Var(X) - Var(Y) = 0$. By the theorem this tells us that $X+Y$ and $X-Y$ are independent. Apparently this only works for iid Normal distributions.

According to Stat 110, the most important MV discrete distribution is the Multinomial. The most important MV continuous distribution is the MVN.