# What should be the output of the pdf for multivariate normal distrubition?

Let's think we have a multivariate normal distribution with $$\mathbf x_1=\begin{bmatrix}x_{11}\\x_{12}\end{bmatrix},\quad \mathbf x_2=\begin{bmatrix}x_{21}\\x_{22}\end{bmatrix},\quad\mathbf m=\begin{bmatrix}m_1\\m_2\end{bmatrix}, \quad\text{and}\quad\mathbf S.$$ Where $\mathbf m$ is the mean, and $\mathbf S$ is a $2\times 2$ covariance matrix.

• What should the result of the PDF be? Should it be a $2\times 1$ array or a single value?
• Can any of its values be bigger than $1$?
• What is $x$, what is $m$ and what is $S$ defining? – Maximilian Matthé Feb 27 '17 at 12:58
• Doesn't this wikipedia article answer your question? And if not, please explain what is still unclear. – Matt L. Feb 27 '17 at 12:58
• @Maximilian is probably feature, m is mean and S is coveriance matrix. – bcan Feb 27 '17 at 13:03
• @Matt Sort of, the thing that confuses my mind is every x has also 2 values. – bcan Feb 27 '17 at 13:04
• Wouldn't that just be a 4-dimensional distribution? Why are the elements $x_{ij}$ grouped in such a way? – Matt L. Feb 27 '17 at 13:08

## 1 Answer

The PDF $p$ of a, say, bivariate Normal distribution with mean $\mu$ and covariance $S$ is a function that takes two parameters and has a scalar-valued output:

$$p: \mathbb{R}^2 \mapsto \mathbb{R}$$

and is given by

$$p(x_1,x_2) = \frac{1}{2\pi |S|}\exp\left(-\frac{1}{2}(\vec{x}-\vec{\mu})^TS^{-1}(\vec{x}-\vec{\mu})\right)$$

where $\vec{\mu}\in\mathbb{R}^2$ is the mean of the distribution and $S\in\mathbb{R}^{2\times 2}$ is its covariance matrix. $\vec{x}=[x_1, x_2]^T$ is the function input and $p(x_1, x_2)$ is the function output.

So, the output is a scalar (a single value). Furthermore, the PDF can become larger than one for some input $\vec{x}$.