Index: The Book of Statistical Proofs ▷ Statistical Models ▷ Multivariate normal data ▷ Multivariate Gaussian ▷ Bivariate normally distributed data

Definition: Bivariate normally distributed data are defined as a set of two-dimensional vectors $y = \left\lbrace y_1, \ldots, y_n \right\rbrace$ with $y_i \in \mathbb{R}^2, \; i = 1,\ldots,n$, independent and identically distributed according to a bivariate normal distribution with mean vector $\mu$ and covariance matrix $\Sigma$

\[\label{eq:bvn} y_i \sim \mathcal{N}(\mu, \Sigma), \quad i = 1, \ldots, n\]

where $\mu$ and $\Sigma$ can be parametrized in terms of means $\mu_1$, $\mu_2$, variances $\sigma_1^2$, $\sigma_2^2$ and correlation $\rho$:

\[\label{eq:bvn-mu-Sigma} \mu = \left[ \begin{matrix} \mu_1 \\ \mu_2 \end{matrix} \right] \quad \text{and} \quad \Sigma = \left[ \begin{matrix} \sigma_1^2 & \rho \sigma_1 \sigma_2 \\ \rho \sigma_1 \sigma_2 & \sigma_2^2 \end{matrix} \right] \; .\]
 
Sources:

Metadata: ID: D222 | shortcut: bvn-data | author: JoramSoch | date: 2025-06-20, 11:44.