2D Random Vector $\boldsymbol{(X, Y)}$ Distribution Experiment
Core Theory
If $X, Y$ are random variables defined on the same probability space, then $(X, Y)$ is called a 2D random vector.
$$F(x, y) = P(X \le x, Y \le y)$$
For continuous random vectors, if there exists a non-negative function $f(x, y)$ such that:
$$F(x, y) = \int_{-\infty}^{y} \int_{-\infty}^{x} f(u, v) \,du\,dv$$
Current Model: Bivariate Normal Distribution
$$f(x,y) = \frac{1}{2\pi\sigma_x\sigma_y\sqrt{1-\rho^2}}
\exp\left[-\frac{1}{2(1-\rho^2)} Q(x,y)\right]$$
where $\rho$ is the correlation coefficient that controls the degree of linear correlation between X and Y.
Joint Probability Density Surface $z = f(x, y)$
Drag to rotate view, adjust parameters to observe the effect of correlation $\rho$
Marginal Distribution and CDF Projection Concepts
● The volume under the surface equals 1
● Cross-sections along axes are marginal probability densities