Multivariate Distributions

Wolfgang Karl Härdle and Zdeněk Hlávka
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Wolfgang Karl Härdle: Humboldt-Universität zu Berlin, C.A.S.E. Centre f. Appl. Stat. & Econ. School of Business and Economics
Zdeněk Hlávka: Charles University in Prague, Faculty of Mathematics and Physics Department of Statistics

Chapter Chapter 4 in Multivariate Statistics, 2015, pp 43-70 from Springer

Abstract: Abstract 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.

Keywords: Probability Density Function; Random Vector; Conditional Distribution; Conditional Expectation; Conditional Variance (search for similar items in EconPapers)
Date: 2015
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DOI: 10.1007/978-3-642-36005-3_4

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