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

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:

\[\label{eq:bvn-pdf} f_X(x) = \frac{1}{2 \pi \sqrt{\sigma_1^2 \sigma_2^2 - \sigma_{12}^2}} \cdot \exp \left[ -\frac{1}{2} \frac{\sigma_2^2 (x_1-\mu_1)^2 - 2 \sigma_{12} (x_1-\mu_1)(x_2-\mu_2) + \sigma_1^2 (x_2-\mu_2)^2}{\sigma_1^2 \sigma_2^2 - \sigma_{12}^2} \right] \; .\]

Proof: 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 $n = 2$, $\mu = \left[ \begin{matrix} \mu_1 \\ \mu_2 \end{matrix} \right]$ and $\Sigma = \left[ \begin{matrix} \sigma_1^2 & \sigma_{12} \\ \sigma_{12} & \sigma_2^2 \end{matrix} \right]$, we obtain:

\[\label{eq:bvn-pdf-s1} \begin{split} f_X(x) &= \frac{1}{\sqrt{(2 \pi)^2 \left| \left[ \begin{matrix} \sigma_1^2 & \sigma_{12} \\ \sigma_{12} & \sigma_2^2 \end{matrix} \right] \right|}} \cdot \exp \left[ -\frac{1}{2} \left( \left[ \begin{matrix} x_1 \\ x_2 \end{matrix} \right] - \left[ \begin{matrix} \mu_1 \\ \mu_2 \end{matrix} \right] \right)^\mathrm{T} \left[ \begin{matrix} \sigma_1^2 & \sigma_{12} \\ \sigma_{12} & \sigma_2^2 \end{matrix} \right]^{-1} \left( \left[ \begin{matrix} x_1 \\ x_2 \end{matrix} \right] - \left[ \begin{matrix} \mu_1 \\ \mu_2 \end{matrix} \right] \right) \right] \\ &= \frac{1}{2 \pi \left| \left[ \begin{matrix} \sigma_1^2 & \sigma_{12} \\ \sigma_{12} & \sigma_2^2 \end{matrix} \right] \right|^\frac{1}{2}} \cdot \exp \left[ -\frac{1}{2} \left[ \begin{matrix} (x_1-\mu_1) & (x_2-\mu_2) \end{matrix} \right] \left[ \begin{matrix} \sigma_1^2 & \sigma_{12} \\ \sigma_{12} & \sigma_2^2 \end{matrix} \right]^{-1} \left[ \begin{matrix} (x_1-\mu_1) \\ (x_2-\mu_2) \end{matrix} \right] \right] \; . \end{split}\]

Using the determinant of a $2 \times 2$ matrix

\[\label{eq:det-2x2} \left| \left[ \begin{matrix} a & b \\ c & d \end{matrix} \right] \right| = a d - b c\]

and the inverse of of a $2 \times 2$ matrix

\[\label{eq:inv-2x2} \left[ \begin{matrix} a & b \\ c & d \end{matrix} \right]^{-1} = \frac{1}{a d - b c} \left[ \begin{matrix} d & -b \\ -c & a \end{matrix} \right] \; ,\]

the probability density function becomes:

\[\label{eq:bvn-pdf-s2} \begin{split} f_X(x) &= \frac{1}{2 \pi \sqrt{\sigma_1^2 \sigma_2^2 - \sigma_{12}^2}} \cdot \exp \left[ -\frac{1}{2 (\sigma_1^2 \sigma_2^2 - \sigma_{12}^2)} \left[ \begin{matrix} (x_1-\mu_1) & (x_2-\mu_2) \end{matrix} \right] \left[ \begin{matrix} \sigma_2^2 & -\sigma_{12} \\ -\sigma_{12} & \sigma_1^2 \end{matrix} \right] \left[ \begin{matrix} (x_1-\mu_1) \\ (x_2-\mu_2) \end{matrix} \right] \right] \\ &= \frac{1}{2 \pi \sqrt{\sigma_1^2 \sigma_2^2 - \sigma_{12}^2}} \cdot \exp \left[ -\frac{1}{2 (\sigma_1^2 \sigma_2^2 - \sigma_{12}^2)} \left[ \begin{matrix} \sigma_2^2 (x_1-\mu_1) - \sigma_{12} (x_2-\mu_2) & \sigma_1^2 (x_2-\mu_2) - \sigma_{12} (x_1-\mu_1) \end{matrix} \right] \left[ \begin{matrix} (x_1-\mu_1) \\ (x_2-\mu_2) \end{matrix} \right] \right] \\ &= \frac{1}{2 \pi \sqrt{\sigma_1^2 \sigma_2^2 - \sigma_{12}^2}} \cdot \exp \left[ -\frac{1}{2 (\sigma_1^2 \sigma_2^2 - \sigma_{12}^2)} (\sigma_2^2 (x_1-\mu_1)^2 - \sigma_{12} (x_1-\mu_1)(x_2-\mu_2) + \sigma_1^2 (x_2-\mu_2)^2 - \sigma_{12} (x_1-\mu_1)(x_2-\mu_2)) \right] \\ &= \frac{1}{2 \pi \sqrt{\sigma_1^2 \sigma_2^2 - \sigma_{12}^2}} \cdot \exp \left[ -\frac{1}{2} \frac{\sigma_2^2 (x_1-\mu_1)^2 - 2 \sigma_{12} (x_1-\mu_1)(x_2-\mu_2) + \sigma_1^2 (x_2-\mu_2)^2}{\sigma_1^2 \sigma_2^2 - \sigma_{12}^2} \right] \; . \end{split}\]
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Metadata: ID: P416 | shortcut: bvn-pdf | author: JoramSoch | date: 2023-09-22, 12:59.