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Probability Theory
Published in Paul L. Goethals, Natalie M. Scala, Daniel T. Bennett, Mathematics in Cyber Research, 2022
for −∞<x<∞. The shape of this density function is the classic bell-shaped curve, symmetric around x=μ. Many natural random phenomena follow this distribution (approximately), and it is also the limiting distribution for several other well-known distributions. Furthermore, a result called the Central Limit Theorem says that the sum of a large number of independent and identically distributed random variables has a distribution that is approximately Normal. This is one of the most remarkable and important results in all of probability theory.
Modeling with Stochastic Differential Equations
Published in Sandip Banerjee, Mathematical Modeling, 2021
Let S be a sample space of the random experiment E. A random variable (or a variate) is a function or mapping X:S→R(set of real numbers), which assigns to each element ω∈S, one and only one number X(ω)=a. The set of all values which X takes, that is, the range of the function X, is called the spectrum of the random variable X, which is a set of real numbers B={a:a=X(ω),ω∈S}. A random variable is called a discrete random variable if it takes on a finite or countably infinite number of values and it is called continuous random variable if it takes on a noncountably infinite number of values.
Reinforcement Learning Problems
Published in Chong Li, Meikang Qiu, Reinforcement Learning for Cyber-Physical Systems, 2019
In the MAB problem, we do not know the true reward distributions for each machine denoted as p(r|a = j): this conditional probability means the probability of receiving reward r given that the agent’s action a was to play the jth machine. If the agent knew these distributions for every slot machine, then it could use them to compute the expected reward for each machine using either Equation (3.2) or (3.3), respectively, according to whether the rewards were continuously or discretely distributed. As such, our bandit must either approximate the expected reward for each machine, 𝔼r|a], or learn the distribution p(r|a). We will focus on the former. Assuming each realization of a random variable is independent and identically distributed (i.i.d.), the law of large numbers states that the empirically estimated expectation approaches the true expectation as the number of samples goes to infinity. That is, () 𝔼[x]=limN→∞1N∑i=1Nxi.(3.4)
A New Ultra-Capacitor Bank Switching Using Probabilistic DC Prediction in Electrical Vehicle
Published in Electric Power Components and Systems, 2023
Ashok Manori, Ashutosh Trivedi, Samta Manori, Jaipal Saroha
Probability Distribution Function (PDF) describes all the possible outcomes that a random variable can possess within a given range of data. There is an upper and a lower bound to these possible outcomes within the given range. However, there are a number of factors on which the precise plotting of these values depends, like, mean, standard deviation, skewness, and kurtosis. In the proposed work the normal distribution has been used. The normal distribution is fully characterized by its mean and standard deviation, i.e., the distribution is not skewed and does exhibit kurtosis. This makes the distribution symmetric and it is depicted as a bell-shaped curve when plotted. A random variable Φ(i) is said to be normally distributed with mean μ and variance σ2 if its PDF is:
Public transport transfers assessment via transferable utility games and Shapley value approximation
Published in Transportmetrica A: Transport Science, 2021
Giorgio Gnecco, Yuval Hadas, Marcello Sanguineti
The approach adopted in this paper is based on the Monte Carlo (MC) method, originally developed for numerical integration (Robert and Casella 2004). It is based on the fact that by the law of large numbers an integral describing the expected value of some random variable can be approximated by taking the empirical mean of independent samples of the variable itself. The possibility to apply the MC method arises since the Shapley value, , can be interpreted as the expected marginal utility provided by the player i to any coalition, when one takes into account all the possible coalitional scenarios, and assumes the same probability for all the possible orders of players. Indeed, the factor in Equation (15) (see Appendix 1) derives from taking into consideration the order of the players when they enter a coalition. In other words, is the expected marginal utility of the player i, when the players enter the coalition according to the order induced by any equally likely permutation p of the sequence. For instance, for , the permutation implies that the players enter the coalition according to the order .
Probability distribution functions of turbulence using multiple criteria over non-uniform sand bed channel
Published in ISH Journal of Hydraulic Engineering, 2020
Probability density function (PDFs) of a random variable is a function that defines the relative possibility for the random variable to deal with a given value. Probability distribution function can be represented by Gaussian distribution, Log normal distribution, and many other etc. These distributions are based on the interval over which the convergence is finite. The advantage of Gram Charlier (GC) series to other series is that it is applicable to series containing interval over which the application of convergence is infinite (Cramer 1999). Also PDFs of GC series is developed considering the higher order moments up to four. The knowledge of PDF of turbulent parameters such as velocity fluctuation and Reynolds shear stress helps to understand the related turbulent parameters (Afzal et al. 2009; Van Atta and Chen 1968).