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Concepts of Probability
Published in Richard Holland, Richard St. John, Statistical Electromagnetics, 2020
Richard Holland, Richard St. John
Inother words, the characteristic or moment-generating function gives immediate access to the moments of the associated distribution. The characteristic function uniquely determines the pdf and vice-versa. In some references, the (−jω) of (2.60) and (2.61) is replaced by (jω) or t; these different-looking formulas have the same interpretation.
Optimal reinsurance and investment problem with default risk and bounded memory
Published in International Journal of Control, 2020
Chao Deng, Wenlong Bian, Baiyi Wu
We assume that, without reinsurance and investment, the reserve process of the insurer follows the classical jump-diffusion process: where x>0 denotes the initial capital of the insurer, c>0 denotes the constant rate of premium received by the insurer, is a standard Brownian motion that is independent of , is a Poisson process with intensity , and the claim sizes is an i.i.d. non-negative random variable sequence with a common continuous distribution . We assume that the claim sizes are independent of the arrival process , and . The moment generating function of is denoted as .
Risk-averse real driving emissions optimization considering stochastic influences
Published in Engineering Optimization, 2020
A. Wasserburger, C. Hametner, N. Didcock
Let X be a continuous random variable on the probability space . Let . The Value-at-Risk (VaR) at the confidence level α is defined as The Conditional Value-at-Risk (CVaR) at the confidence level α is defined as The Entropic Value-at-Risk (EVaR) at the confidence level α is defined as where is the moment-generating function.
A generalized alarm delay-timer’s performance indices computing method
Published in Systems Science & Control Engineering, 2018
Jingqiu Lin, Donghao Sun, Xingzhen Bai, Lida Chen
The calculation of the average alarm delay is based on the Markov model in Figure 4. According to the definition in (3), the AAD for out of samples generalized alarm delay-timer is where is the number of steps taken from 1 to . is the moment generating function of the discrete random variable , (Papoulis & Saunders, 1989), is the sampling period.