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Extra resources for Advanced Stochastic Models, Risk Assessment, and Portfolio Optimization: The Ideal Risk, Uncertainty, and Performance Measures
The value can be obtained by calculating the volume under the density function f . Because there are n random variables, we have now n arguments for both functions: the density function and the cumulative distribution function. It is also possible to express the density function in terms of the distribution function by computing sequentially the first-order partial derivatives of the distribution function with respect to all variables, fX (x1 , . . 3 ∂ n FX (x1 , . . , xn ) . ∂x1 . . 2) Marginal Distributions Beside this joint distribution, we can consider the above mentioned marginal distributions, that is, the distribution of one single random variable Xi .
17 Elliptical 16 The reader interested in outdoor activities such as hiking or climbing as well as geographically interested people might know the concept of level curves from their hiking maps, where the mountains are visualized by there iso-level lines. 17 For a thorough introduction into the class of ellipitcal distribution, see Fang et al. (1994). 25 Concepts of Probability distributions with existing density function can be described by a triple (µ, , g),18 where µ and play similar roles as the mean vector and the variance-covariance matrix in the multivariate normal setting.
4 Student’s t -distribution Student’s t-distributions are used in finance as probabilistic models of assets returns. The density function of the t-distribution is given by the following equation: x2 1 ((n + 1)/2) fX (x) = √ 1+ (n/2) n πn 1 − n+ 2 , x ∈ R, 12 ADVANCED STOCHASTIC MODELS where n is an integer valued parameter called degree of freedom. For large values of n, the t-distribution doesn’t significantly differ from a standard normal distribution. Usually, for values n > 30, the t-distribution is considered as equal to the standard normal distribution.