D2L - Chapter 2.6 Probablity and Statistics + 2.7 Documentation

Probablity and Statistics

Covariance Matrix, Diagonal & Off-Diagonal Entries

1. Covariance Matrix
   A covariance matrix \(\Sigma\) is a square matrix that shows the covariance between each pair of coordinates in a random vector. It is:

   • Symmetric
   • Positive semidefinite

2. Diagonal Entries

   • Each diagonal entry \(\Sigma\_{ii}\) is the variance of the \(i\)-th coordinate, denoted \(\mathrm{Var}(x_i)\).
   • Variance measures how spread out the \(i\)-th coordinate is around its mean.

3. Off-Diagonal Entries

   • Each off-diagonal entry \(\Sigma\_{ij}\) (with \(i \neq j\)) is the covariance \(\mathrm{Cov}(x_i, x_j)\).
   • Positive covariance means the two coordinates tend to vary in the same direction (above/below their means together), negative covariance indicates they move in opposite directions, and zero covariance indicates no linear co-movement.

4. Why Symmetric?

   • Because \(\mathrm{Cov}(x_i, x_j) = \mathrm{Cov}(x_j, x_i)\), the matrix \(\Sigma\) must be symmetric.

Documentation

print(dir(jax.random))
help(jax.numpy.ones)