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Multivariate Gaussian Distribution

This is the 2D or higher dimension version of 20260120T172840-gaussian_distribution

  • Assume we have two random variable $X, Y$
  • Let them be normally distributed on their own
  • If we have a function $f(x, y)$ that represented the probability of both of them occurring together we have sort of have a 2D PDF of the two variables
  • They form a gaussian surface in $\mathbb{R}^3$

How would we describe this?

  • In 1-D we only need two numbers - mean and standard deviation
  • But in 2D we have 4 quantities
    • Mean vector
    • Variance of X
    • Variance of Y
    • Co-variance of X, Y
  • Concisely we can represent the variances as a covariance matrix

$$ \Sigma = \begin{bmatrix} \mathrm{var}(X) & \mathrm{cov}(X, Y)\\ \mathrm{cov}(X, Y) & \mathrm{Var}(Y) \end{bmatrix} $$

  • We can denote the mean vector as $\pmb{\mu}$

$\mathcal{N}(\pmb{\mu}, \Sigma)$

This is still very similar to our 1D example. The mean here is a mean vector and the covariance matrix is just the variance of a vector in $\mathbb{R}^2jjjjjjjjjjjjj:w $