Multivariate Distributions

A random vector Random vector is a Cumulative distribution function cdf vector of cdfCumulative distribution function (cdf) random variables. A random vector $$X \in \mathbb{R}^{p}$$ has pdfProbability density function (pdf) a multivariate cumulative distribution function (cdf) and a multivariate probability density function (pdf). DensityProbability density function (pdf) Distribution functionCumulative distribution function (cdf) Probability density function (pdf) Cumulative distribution function cdf They are defined as: $$\displaystyle\begin{array}{rcl} F_{X}(x)& =& \mathop{\mathrm{\mathsf{P}}}\nolimits (X \leq x) {}\\ & =& \mathop{\mathrm{\mathsf{P}}}\nolimits (X_{1} \leq x_{1},X_{2} \leq x_{2},\ldots,X_{p} \leq x_{p}) {}\\ & =& \int \limits _{-\infty }^{\infty }\ldots \int \limits _{ -\infty }^{\infty }f_{ X}(x_{1},x_{2},\ldots,x_{p})dx_{1}dx_{2}\ldots dx_{p}, {}\\ \end{array}$$ and if the cdf F X (. ) is differentiable, the pdf f X (. ) is $$\displaystyle\begin{array}{rcl} f_{X}(x)& =& \frac{\partial ^{p}F(x)} {\partial x_{1}\ldots \partial x_{p}}. {}\\ \end{array}$$ Important features that can be extracted from F X (. ) and f X (. ) are the mutual dependencies of the elements of X, moments, and multivariate tail behavior.