\(\begin{bmatrix}x_i \\ y_i\end{bmatrix} \sim \mathcal{N}\left(
\begin{bmatrix}\mu_x \\ \mu_y \end{bmatrix},
\mathbf{\Sigma} = \begin{bmatrix}\sigma_x^2 & \rho \sigma_x \sigma_y \\
\rho \sigma_y \sigma_x & \sigma_y^2
\end{bmatrix}
\right) \)
Pairs of observations \(x_i, y_i\) are drawn from a bivariate normal distribution with means \(\mu_x, \mu_y\) and variance-covariance matrix \(\mathbf{\Sigma}\).
- \(\rho\)
- "rho", correlation coefficient
- \(\sigma_x\)
- "sigma x", standard deviation of x
- \(\sigma_y\)
- "sigma y", standard deviation of y
- \(\mu_x\)
- "mu x", mean of x
- \(\mu_y\)
- "mu y", mean of y
- plot
- darker rings = greater probability; points are 100 random x,y pairs
- math
- math behind the simulation
- code
- R code for simulation
- \(\rho\)
- "rho", correlation coefficient
- \(\sigma_x\)
- "sigma x", standard deviation of x
- \(\sigma_y\)
- "sigma y", standard deviation of y
- \(\mu_x\)
- "mu x", mean of x
- \(\mu_y\)
- "mu y", mean of y
