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Additional resources for Advanced Stochastic Models, Risk Assessment, and Portfolio Optimization: The Ideal Risk, Uncertainty, and Performance Measures
In such cases, the analysis may require us to resort to general arguments based on certain general inequalities from the theory of probability. In this section, we give an account of such inequalities and provide illustration where possible. 1 Chebyshev’s Inequality Chebyshev’s inequality provides a way to estimate the approximate probability of deviation of a random variable from its mean. Its most simple form concerns positive random variables. Suppose that X is a positive random variable, X > 0.
Simply speaking, an n-dimensional random vector X with density function f is spherically distributed if all the level curves,16 that is, the set of all points where the density function f admits a certain value c, possesses the form of a sphere. In the special case when n = 2, the density function can be plotted and the level curves look like circles. Analogously, a n-dimensional random vector X with density function f is elliptically distributed if the form of all level curves equals the one of an ellipse.
Xn )dx1 . . dxi − 1 dxi + 1 . . dxn Dependence of Random Variables Typically, when considering multivariate distributions, we are faced with inference between the distributions; that is, large values of one random variable imply large values of another random variable or small values of a third random variable. S. citizen, and X2 , the weight of this citizen, then large values of X1 tend to result in large values of X2 . This property is denoted as the dependence of random variables and a powerful concept to measure dependence will be introduced in a later section on copulas.