Let $Y$ ~ $MVN_3(\mu, \Sigma)$ where $\mu = (5,6,7)$ and
$\Sigma = \begin{bmatrix}2 & 0 & 1\\0 & 3 & 2\\1&2&4\end{bmatrix}$
Find
(a) The marginal distribution of $Y_1$
(b) The joint distribution of $Y_3$ and $Y_1$
(c) The conditional distribution of $Y_3$ given $Y_1$.
Now, my experience with probability prior to this class is limited to Sheldon Ross's intro book and Wackerley, Mendenhall, and Schaeffer, which I'm currently using for my Into Math Stat class. I've never before done conditional, joint, or marginal densities with matrices, nor have I covered them for variables that are dependent…. So please forgive the embarassing attempts I have made to solve this. It also does not help that the class this problem is for does not have a textbook, just some notes frantically scrambled onto a whiteboard twice a week while our prof reminds us that he hates teaching and would rather be researching…
My attempts
(a) I can write
$Y_1 = AY + b$, where $b=(0,0,0)^T$ and $A = (1,0,0)^T$.
Thus: $Y_1$~$MVN(\cdot,\cdot)$ because it is a linear transformation of a multivariate normal.
Thus: $E[Y_1]=E[AY+b]=AE[Y]+b=(1,0,0)^T(5,6,7)^T=5$
$Cov(AY) = ACov(Y)A^T = A\begin{bmatrix}2 & 0 & 1\\0 & 3 & 2\\1&2&4\end{bmatrix}A^T$
This becomes: $(2,0,1)\begin{bmatrix}1\\0\\0\end{bmatrix}=2$
However, knowing that $Y_1$ and $Y_3$ are not independent makes me question this result…
(b) The joint distribution of $Y_1$ and $Y_3$
I basically did the same thing as above but with $A = \begin{bmatrix}1 & 0 & 0\\0&0&1\end{bmatrix}$.
This yielded $W = (Y_1, Y_3)^T$ ~ $MVN(\mu=(5,7)^T, \Sigma = \begin{bmatrix}2 & 1\\1&4\end{bmatrix})$
(c) The conditional distribution of $Y_3$ given $Y_1$
This part I'm honestly at a loss for…unless I'm just supposed to take the integral of the density function for $(Y_1, Y_3)$ divided by the density function for $(Y_1)$, i.e., conditional = $\int f_{Y_1,Y_3}/f_{Y_1} dy_1$.
If I am supposed to do this, how do I get the actual joint density function and perform the integration and division?
The only formulas I've been provided for multivariate normal distributions involve matrices, and I have never seen matrix integration. I apologize if my comprehension is lacking, my classmates and I have been thrown into this material with no assistance from our professor.
Best Answer
Alrighty, y'all. I have an answer. Sorry it took me so long to get it posted here. School was absolutely hectic this week. Spring break is here, though, and I can type up my answer.
First we need to find the joint distribution of $(Y_1, Y_3)$. Since $Y\sim MVN( \mu, \Sigma)$ we know that any subset of the components of $Y$ is also $MVN$. Thus we use
$$ A = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ \end{pmatrix} $$ And see that
$$ AY = (Y_1, Y_3)^T $$ $$ \Sigma = \begin{pmatrix} 2 & 1 \\ 1 & 4 \\ \end{pmatrix} $$ $$ \mu(Y_1,Y_2) = (5,7)^T $$ Therefore, using the theorem for conditional distributions of a multivariate normal yields:
$$\begin{align}\newcommand{\c}{\text{Cov}} \newcommand{\v}{\text{Var}} E[Y_3|Y_1]&=μ_{Y_3}+\frac{\c(Y_1,Y_3)(Y_1−μ_{Y_1})}{\v(Y_1)}\\ &=\frac{9+Y_1}{2} \end{align}$$ And
$$\begin{align} \v(Y_3|Y_1) &= \v(Y_3) - \frac{\c(Y_1,Y_3)^2}{\v(Y_1)} \\ &= 4 - \frac{1}{2} = \frac{7}{2} \end{align}$$