Index: The Book of Statistical Proofs ▷ Probability Distributions ▷ Multivariate continuous distributions ▷ Multivariate normal distribution ▷ Probability density function in terms of correlation coefficient

Theorem: Let $X = \left[ \begin{matrix} X_1 \\ X_2 \end{matrix} \right]$ follow a bivariate normal distribution:

\[\label{eq:bvn} X \sim \mathcal{N}\left( \mu = \left[ \begin{matrix} \mu_1 \\ \mu_2 \end{matrix} \right], \Sigma = \left[ \begin{matrix} \sigma_1^2 & \sigma_{12} \\ \sigma_{12} & \sigma_2^2 \end{matrix} \right] \right) \; .\]

Then, the probability density function of $X$ is

\[\label{eq:bvn-pdf} f_X(x) = \frac{1}{2 \pi \, \sigma_1 \sigma_2 \sqrt{1 - \rho^2}} \cdot \exp \left[ -\frac{1}{2 (1 - \rho^2)} \left( \left( \frac{x_1-\mu_1}{\sigma_1} \right)^2 - 2 \rho \frac{(x_1-\mu_1) (x_2-\mu_2)}{\sigma_1 \sigma_2} + \left( \frac{x_2-\mu_2}{\sigma_2} \right)^2 \right) \right]\]

where $\rho$ is the correlation between $X_1$ and $X_2$.

Proof: Since $X$ follows a special case of the multivariate normal distribution, its covariance matrix is

\[\label{eq:cov-X} \mathrm{Cov}(X) = \Sigma = \left[ \begin{matrix} \sigma_1^2 & \sigma_{12} \\ \sigma_{12} & \sigma_2^2 \end{matrix} \right]\]

and the covariance matrix can be decomposed into correlation matrix and standard deviations:

\[\label{eq:Sigma} \begin{split} \Sigma &= \left[ \begin{matrix} \sigma_1^2 & \rho \, \sigma_1 \sigma_2 \\ \rho \, \sigma_1 \sigma_2 & \sigma_2^2 \end{matrix} \right] \\ &= \left[ \begin{matrix} \sigma_1 & 0 \\ 0 & \sigma_2 \end{matrix} \right] \left[ \begin{matrix} 1 & \rho \\ \rho & 1 \end{matrix} \right] \left[ \begin{matrix} \sigma_1 & 0 \\ 0 & \sigma_2 \end{matrix} \right] \; . \end{split}\]

The determinant of this matrix is

\[\label{eq:Sigma-det} \begin{split} \left| \Sigma \right| &= \left| \left[ \begin{matrix} \sigma_1 & 0 \\ 0 & \sigma_2 \end{matrix} \right] \left[ \begin{matrix} 1 & \rho \\ \rho & 1 \end{matrix} \right] \left[ \begin{matrix} \sigma_1 & 0 \\ 0 & \sigma_2 \end{matrix} \right] \right| \\ &= \left| \left[ \begin{matrix} \sigma_1 & 0 \\ 0 & \sigma_2 \end{matrix} \right] \right| \cdot \left| \left[ \begin{matrix} 1 & \rho \\ \rho & 1 \end{matrix} \right] \right| \cdot \left| \left[ \begin{matrix} \sigma_1 & 0 \\ 0 & \sigma_2 \end{matrix} \right] \right| \\ &= (\sigma_1 \sigma_2) (1 - \rho^2) (\sigma_1 \sigma_2) \\ &= \sigma_1^2 \sigma_2^2 (1 - \rho^2) \end{split}\]

and the inverse of this matrix is

\[\label{eq:Sigma-inv} \begin{split} \Sigma^{-1} &= \left( \left[ \begin{matrix} \sigma_1 & 0 \\ 0 & \sigma_2 \end{matrix} \right] \left[ \begin{matrix} 1 & \rho \\ \rho & 1 \end{matrix} \right] \left[ \begin{matrix} \sigma_1 & 0 \\ 0 & \sigma_2 \end{matrix} \right] \right)^{-1} \\ &= \left[ \begin{matrix} \sigma_1 & 0 \\ 0 & \sigma_2 \end{matrix} \right]^{-1} \left[ \begin{matrix} 1 & \rho \\ \rho & 1 \end{matrix} \right]^{-1} \left[ \begin{matrix} \sigma_1 & 0 \\ 0 & \sigma_2 \end{matrix} \right]^{-1} \\ &= \frac{1}{1 - \rho^2} \left[ \begin{matrix} 1/\sigma_1 & 0 \\ 0 & 1/\sigma_2 \end{matrix} \right] \left[ \begin{matrix} 1 & -\rho \\ -\rho & 1 \end{matrix} \right] \left[ \begin{matrix} 1/\sigma_1 & 0 \\ 0 & 1/\sigma_2 \end{matrix} \right] \; . \end{split}\]

The probability density function of the multivariate normal distribution for an $n \times 1$ random vector $x$ is:

\[\label{eq:mvn-pdf} f_X(x) = \frac{1}{\sqrt{(2 \pi)^n |\Sigma|}} \cdot \exp \left[ -\frac{1}{2} (x-\mu)^\mathrm{T} \Sigma^{-1} (x-\mu) \right] \; .\]

Plugging in $n = 2$, $\mu$ from \eqref{eq:bvn} and $\Sigma$ from \eqref{eq:Sigma-det} and \eqref{eq:Sigma-inv}, the probability density function becomes:

\[\label{eq:bvn-pdf-corr} \begin{split} f_X(x) &= \frac{1}{\sqrt{(2 \pi)^2 \sigma_1^2 \sigma_2^2 (1 - \rho^2)}} \cdot \exp \left[ -\frac{1}{2} \left( \left[ \begin{matrix} x_1 \\ x_2 \end{matrix} \right] - \left[ \begin{matrix} \mu_1 \\ \mu_2 \end{matrix} \right] \right)^\mathrm{T} \frac{1}{1 - \rho^2} \left[ \begin{matrix} 1/\sigma_1 & 0 \\ 0 & 1/\sigma_2 \end{matrix} \right] \left[ \begin{matrix} 1 & -\rho \\ -\rho & 1 \end{matrix} \right] \left[ \begin{matrix} 1/\sigma_1 & 0 \\ 0 & 1/\sigma_2 \end{matrix} \right] \left( \left[ \begin{matrix} x_1 \\ x_2 \end{matrix} \right] - \left[ \begin{matrix} \mu_1 \\ \mu_2 \end{matrix} \right] \right) \right] \\ &= \frac{1}{2 \pi \, \sigma_1 \sigma_2 \sqrt{1 - \rho^2}} \cdot \exp \left[ -\frac{1}{2 (1 - \rho^2)} \left[ \begin{matrix} \frac{x_1-\mu_1}{\sigma_1} & \frac{x_2-\mu_2}{\sigma_2} \end{matrix} \right] \left[ \begin{matrix} 1 & -\rho \\ -\rho & 1 \end{matrix} \right] \left[ \begin{matrix} \frac{x_1-\mu_1}{\sigma_1} \\ \frac{x_2-\mu_2}{\sigma_2} \end{matrix} \right] \right] \\ &= \frac{1}{2 \pi \, \sigma_1 \sigma_2 \sqrt{1 - \rho^2}} \cdot \exp \left[ -\frac{1}{2 (1 - \rho^2)} \left[ \begin{matrix} \left( \frac{x_1-\mu_1}{\sigma_1} - \rho \frac{x_2-\mu_2}{\sigma_2} \right) & \left( \frac{x_2-\mu_2}{\sigma_2} - \rho \frac{x_1-\mu_1}{\sigma_1} \right) \end{matrix} \right] \left[ \begin{matrix} \frac{x_1-\mu_1}{\sigma_1} \\ \frac{x_2-\mu_2}{\sigma_2} \end{matrix} \right] \right] \\ &= \frac{1}{2 \pi \, \sigma_1 \sigma_2 \sqrt{1 - \rho^2}} \cdot \exp \left[ -\frac{1}{2 (1 - \rho^2)} \left( \left( \frac{x_1-\mu_1}{\sigma_1} \right)^2 - 2 \rho \frac{(x_1-\mu_1) (x_2-\mu_2)}{\sigma_1 \sigma_2} + \left( \frac{x_2-\mu_2}{\sigma_2} \right)^2 \right) \right] \; . \end{split}\]
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Metadata: ID: P417 | shortcut: bvn-pdfcorr | author: JoramSoch | date: 2023-09-29, 09:45.