Definition: Bivariate normal distribution
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The Book of Statistical Proofs ▷
Probability Distributions ▷
Multivariate continuous distributions ▷
Bivariate normal distribution ▷
Definition
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Metadata: ID: D189 | shortcut: bvn | author: JoramSoch | date: 2023-09-22, 10:56.
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 $\mu_1$ and $\mu_2$, variances $\sigma_1^2$ and $\sigma_2^2$ and covariance $\sigma_{12}$:
\[\label{eq:bvn} \mu = \left[ \begin{matrix} \mu_1 \\ \mu_2 \end{matrix} \right] \quad \text{and} \quad \Sigma = \left[ \begin{matrix} \sigma_1^2 & \sigma_{12} \\ \sigma_{12} & \sigma_2^2 \end{matrix} \right] \; .\]- Wikipedia (2023): "Multivariate normal distribution"; in: Wikipedia, the free encyclopedia, retrieved on 2023-09-22; URL: https://en.wikipedia.org/wiki/Multivariate_normal_distribution#Bivariate_case.
Metadata: ID: D189 | shortcut: bvn | author: JoramSoch | date: 2023-09-22, 10:56.