Index: The Book of Statistical Proofs ▷ Probability Distributions ▷ Multivariate continuous distributions ▷ Multivariate normal distribution ▷ Bivariate normal distribution

Definition: Let $X$ be an $2 \times 1$ random vector. Then, $X$ is said to have a bivariate normal distribution, if $X$ follows a multivariate normal distribution

\[\label{eq:mvn} X \sim \mathcal{N}(\mu, \Sigma)\]

with means $x_1$ and $x_2$, variances $\sigma_1^2$ and $\sigma_2^2$ and covariance $\sigma_{12}$:

\[\label{eq:bvn} \mu = \left[ \begin{matrix} x_1 \\ x_2 \end{matrix} \right] \quad \text{and} \quad \Sigma = \left[ \begin{matrix} \sigma_1^2 & \sigma_{12} \\ \sigma_{12} & \sigma_2^2 \end{matrix} \right] \; .\]
 
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Metadata: ID: D189 | shortcut: bvn | author: JoramSoch | date: 2023-09-22, 10:56.