Will someone please explain multivariate normal distributions with a real-life example? I understand a concept best when I see it being applied in the real world.
 A: I'll give you an example from Data Communication theory. The input/output relations of a narrowband, single-user MIMO link (multi-input multi-output) can be represented by the complex baseband vector notation $\mathbf{y} = \mathbf{H}\mathbf{x} + \mathbf{n}$, where $\mathbf{x}$ is the $(n_T ×1)$ transmit vector, $\mathbf{y}$ is the $(n_R×1)$ receive vector, $\mathbf{H}$ is the $(n_R × n_T)$ channel matrix, and $\mathbf{n}$ is the $(n_R × 1)$ additive white Gaussian noise (AWGN) vector at a given instant in time.
A general entry of the channel matrix is denoted by ${h_{ij}}$. This represents the complex gain of the channel between the $j$th transmitter and the $i$th receiver. With a MIMO system consisting of $n_T$ transmit antennas and $n_R$ receive antennas, the channel matrix is written as
\begin{equation}
\mathbf{H} = \left[
\begin{matrix}
    h_{11} & h_{12} & h_{13} & \dots  & h_{1n_T} \\
    h_{21} & h_{22} & h_{23} & \dots  & h_{2n_T} \\
    \vdots & \vdots & \vdots & \ddots & \vdots \\
    h_{n_R1} & h_{n_R2} & h_{n_R3} & \dots  & h_{n_Rn_T}
\end{matrix} \right]
\end{equation}

where $h_{ij} = \alpha + j\beta = \left|h_{ij}\right| e^{\phi_{ij}}$.
In a rich scattering environment with no line-of-sight (LOS), the channel gains $\left|h_{ij}\right|$ are usually Rayleigh distributed. If $\alpha$ and $\beta$ are independent and normal distributed random variables, then $\left|h_{ij}\right|$ is a Rayleigh distributed random variable.

Now the question is how much information can be reliably exchanged through this system. The answer is 
\begin{equation}
I = \log_2\left[\det\left( \mathbf{H} \mathbf{\Phi} \mathbf{H}^* \left( \mathbf{K}^n\right)^{-1} + \mathbf{I}_{n_R} \right)  \right]
\end{equation}
where $E \left\{ \mathbf{y}\mathbf{y}^* \right\} = \mathbf{K}$
I'm not going into more details but just mentioning that the multivariate normal distribution is the underlying assumption to develop a very rich subject.
A: Here is an introduction to a course chapter I found online. Hope this helps!:
"Let the random variable Y denote the weight of a randomly selected individual, in pounds. Then, suppose we are interested in determining the probability that a randomly selected individual weighs between 140 and 160 pounds. That is, what is P(140 < Y < 160)?
But, if we think about it, we could imagine that the weight of an individual increases (linearly?) as height increases. If that's the case, in calculating the probability that a randomly selected individual weighs between 140 and 160 pounds, we might find it more informative to first take into account a person's height, say X. That is, we might want to find instead $P(140 \le Y \le 160 \mid X = x)$. To calculate such a conditional probability, we clearly first need to find the conditional distribution of Y given X = x. That's what we'll do in this lesson, that is, after first making a few assumptions.
First, we'll assume that (1) Y follows a normal distribution, (2) $E(Y \mid x)$, the conditional mean of Y given x is linear in x, and (3) $Var(Y \mid x)$, the conditional variance of Y given x is constant. Based on these three stated assumptions, we'll find the conditional distribution of Y given $X = x$. 
Then, to the three assumptions we've already made, we'll then add the assumption that the random variable X follows a normal distribution, too. Based on the now four stated assumptions, we'll find the joint probability density function of X and Y.
Objectives
To find the conditional distribution of Y given X = x, assuming that (1) Y follows a normal distribution, (2) $E(Y \mid x)$, the conditional mean of Y given x is linear in x, and (3) $Var(Y \mid x)$, the conditional variance of Y given x is constant. 
To learn how to calculate conditional probabilities using the resulting conditional distribution.
To find the joint distribution of X and Y, assuming that (1) X follows a normal distribution, (2) Y follows a normal distribution, (3) $E(Y \mid x)$, the conditional mean of Y given x is linear in x, and (4) $Var(Y \mid x)$, the conditional variance of Y given x is constant."
Note this is for bi-variate normal distributions. You can add more variables that approx. normally distributed.
