Math and science::INF ML AI

# Covariance matrix

Let $$X$$ and $$Y$$ be two random variables. The covariance between $$X$$ and $$Y$$ is defined as:

\begin{aligned} Cov[X,Y] &:= E[(X-E[X])(Y-E[Y])] \\ &= [...] \end{aligned}

Let the vector $$Z$$ be defined like so: $$Z := \begin{bmatrix} X \\ Y\end{bmatrix}$$. Thus, $$Z$$ is a vector of random variables.

The covariance matrix for $$Z$$ is defined as:

\begin{aligned} Cov[Z] &:= E[(Z - E[Z])(Z - E[Z])^T] \\ &= [...] \\ \end{aligned}

Where the expectation is an elementwise operation. The covariance matrix is a result of a matrix multiplication of two vector-like matrices, which produces a 2x2 matrix. (Yes, it is valid!).