The true random variables are continuous. But I only have discrete probability distributions as estimation of the continuous random variables.
The following are known.
- $P(X_{i}=x_{i})$
- $P(X_{i}=x_{i}, X_{j}=x_{j})$
Suppose the overall joint probability $P(\vec{X} = \vec{x})$ is a multivariate normal distribution.
How to estimate the overall joint probability as a discrete probability distribution?
From wikipedia:
$\boldsymbol\mu$ is the vector of means.
$\boldsymbol\Sigma$ is the covariance matrix.
$$\begin{align} f_{\mathbf X}(x_1,\ldots,x_k) & = \frac{\exp\left(-\frac 1 2 ({\mathbf x}-{\boldsymbol\mu})^\mathrm{T}{\boldsymbol\Sigma}^{-1}({\mathbf x}-{\boldsymbol\mu})\right)}{\sqrt{(2\pi)^k|\boldsymbol\Sigma|}} \end{align}$$