China
Africa
Explore chapters and articles related to this topic
Some Probability Concepts for Engineers
Published in Richard L. Shell, Ernest L. Hall, Handbook of Industrial Automation, 2000
One of the main applications of the characteristic function is to obtain the central moments of the corresponding random variable. In fact, in we differentiate the characteristic function k times with respect to t, we get ψX(k)(t)=∫−∞∞ikxkeitxf(x)dx
Probability, Random Variables, and Stochastic Processes
Published in Erchin Serpedin, Thomas Chen, Dinesh Rajan, Mathematical Foundations for SIGNAL PROCESSING, COMMUNICATIONS, AND NETWORKING, 2012
The characteristic function can be used to easily compute the moments of the random variable. Using the Taylors series expansion of ejωx, we can expand the characteristic function as () ϕχ(ω)=E{ejωχ} () =E{1+jωx+(jωx)22!+…} () =1+jωE{χ}+(jω)22!E{χ2}+…
Phenomenological Creep Models of Fibrous Composites (Probabilistic Approach)
Published in Leo Razdolsky, Phenomenological Creep Models of Composites and Nanomaterials, 2019
In probability theory and statistics, the characteristic function of any realvalued random variable completely defines its probability distribution. If a random variable admits a probability density function, then the characteristic function is the Inverse Fourier transform of the probability density function. Thus it provides the basis of an alternative route to analytical results compared with working directly with probability density functions or cumulative distribution functions. There are particularly simple results for the characteristic functions of distributions defined by the weighted sums of random variables. The characteristic function always exists when treated as a function of a real-valued argument, unlike the moment-generating function. There are relations between the behavior of the characteristic function of a distribution and properties of the distribution, such as the existence of moments and the existence of a density function. The characteristic function provides an alternative way for describing a random variable. The characteristic function 9X(t) = E[eltX] also completely determines behavior and properties of the probability distribution of the random variable X. The two approaches are equivalent in the sense that by knowing one of the functions it is always possible to find the other, yet they both provide different insight for understanding the features of the random variable. Note however that the characteristic function of a distribution always exists, even when the probability density function or moment-generating function does not. The characteristic function approach is particularly useful in analysis of linear combinations of independent random variables: a classical proof of the Central Limit Theorem uses characteristic functions. Another important application is to the theory of the decomposability of random variables. For a scalar random variable x the characteristic function is defined as the expected value of eitX, where i is the imaginary unit, and t e R is the argument of the characteristic function. If random variable X has a probability density function fX, then the characteristic function is its Fourier transform [21]. Extensive tables of characteristic functions are provided in [22].
Supply chain delivery performance improvement: a white-box perspective
Published in International Journal of Production Research, 2023
Liangyan Tao, Ailin Liang, Maxim A. Bushuev
CF-GERT is an extension of the typical GERT network model. The core idea of CF-GERT (Tao et al. 2017) is to replace the moment generating function (MGF) of GERT with a characteristic function. Then the expected delivery time, variance, etc. are obtained by using the properties of the characteristic function. More importantly, the Cumulative Distribution Function (CDF) and the Probability Density Function (PDF) can be obtained via the inverse Fourier transform of the characteristic function. The following are some key definitions and theorems of CF-GERT, please refer to Tao et al. (2017) for more information.