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Common Statistical Approach
Published in Atsushi Kawaguchi, Multivariate Analysis for Neuroimaging Data, 2021
An appropriate distribution can be considered to be a null distribution. The binomial distribution is often used in the analysis of binary data, but the subjects are independent of each other and the variance is determined by the sample size and the probability of occurrence. The binomial distribution is also approximated by a normal distribution when the sample size is large. The normal distribution is often used for continuous data. If the sample size of data is sufficient, the sample mean follows a normal distribution; otherwise, the data itself must follow a normal distribution. If the variance is known, it follows the normal distribution, but if it is unknown, it follows the t distribution. Using theoretical distributions like these requires assumptions about the data. It is difficult to verify rigorously the assumption and accuracy of data needed to use the method with confidence. Additionally, the difficulty is expected to increase for data with many variables as in multivariate analysis and in multiple comparison.
Phenomenological Creep Models of Fibrous Composites (Probabilistic Approach)
Published in Leo Razdolsky, Phenomenological Creep Models of Composites and Nanomaterials, 2019
P(X) gives the probability of successes in n binomial trials. If p is the probability of success and q is the probability of failure in a binomial trial, then the expected number of successes in n trials (i.e., the mean value of the binomial distribution) is E(X) = μ = np. The variance of the binomial distribution is D(X) = σ2 = npq. Note: In a binomial distribution, only 2 parameters, namely n and p are needed to determine the probability. For the special case where r is an integer, the binomial distribution is known as the Pascal distribution. It is the probability distribution of a certain number of failures and successes in a series of independent and identically distributed Bernoulli trials. For k + r Bernoulli trials with success probability p, the binomial distribution gives the probability of k successes and r failures, with a failure on the last trial. In other words, this type of binomial distribution is the probability distribution of the number of successes before the rth failure in a Bernoulli process, with probability p of successes on each trial. A Bernoulli process is a discrete time process, and so the number of trials, failures, and successes are integers. Consider the following example.
Visualizing Categorical Data
Published in Wendy L. Martinez, Angel R. Martinez, Jeffrey L. Solka, Exploratory Data Analysis with MATLAB®, 2017
Wendy L. Martinez, Angel R. Martinez, Jeffrey L. Solka
We can repeat the experiment for n trials, where each trial is independent and results in a success with probability p. If a variable X denotes the number of successes in the n trials, then we say it follows a binomial distribution with parameters n and p. The binomial probability mass function is P(X=x)=(χn)fχ(1-p)n-x;x=0,1,2…,n;0≤p≤1. $$ P(X = x) = (_{\chi }^{n} )_{{f^{\chi } (1}} - p)^{{n - x}} ;\,x = \,0,1,2 \ldots ,n;\,0 \le p\, \le 1. $$
AGV dispatching algorithm based on deep Q-network in CNC machines environment
Published in International Journal of Computer Integrated Manufacturing, 2021
Kyuchang Chang, Seung Hwan Park, Jun-Geol Baek
A binomial distribution was used to implement the stochastic work execution in each CNC (Holloway and Nelson 1974). Binomial distribution is frequently used to model the number of successes drawn with replacement from a population. In this simulation environment, the binomial distribution with parameters 20 and p is the discrete probability distribution of the number of initiations in a sequence of 20 independent experiments, where 20 is the number of CNCs and p is the hyper-parameter of the experiment. A CNC that starts the work at every step is determined probabilistically by a binomial distribution. The reason for using binomial distribution is not to simulate the actual process as it is; rather, it is to create a variety of situations probabilistically. In a virtual environment with randomness, virtually all the possible states can be generated. Therefore, compared to storing and using actual manufacturing process data, RL can be trained for a wider variety of situations.
Experimental repetitions and blockage of large stems at ogee crested spillways with piers
Published in Journal of Hydraulic Research, 2019
Paloma Furlan, Michael Pfister, Jorge Matos, Conceição Amado, Anton J. Schleiss
The Wald method is the most common approach for calculating symmetric binomial confidence intervals, and is based on the approximation of the binomial by the Normal distribution. If X is binomially distributed with parameters n and π, then X has the same distribution as the sum of n independent Bernoulli random variables (Montgomery & Runger, 2011; Ross, 2010). Then, by the central limit theorem, the binomial distribution can be approximated using a standard Normal distribution as n approaches . A rule of thumb to well-approximate a binomial by a normal can be given by the relation: (Ross, 2010). The Wald interval estimator of π was calculated according to Eq. (2): where denotes the quantile of the standard Normal distribution (Agresti & Coull, 1998; Vollset, 1993). The following term, is called the standard error (SE) of the point estimator .
Partial sums of analytic functions defined by binomial distribution and negative binomial distribution
Published in Applied Mathematics in Science and Engineering, 2022
Rubab Nawaz, Saira Zainab, Fairouz Tchier, Qin Xin, Afis Saliu, Sarfraz Nawaz Malik
One of the most essential discrete probability distributions is Binomial distribution. When there are two possible outcomes, then Binomial distribution model is used which is an important probability model. In a Bernoulli trial, the random experiment has two hypothetical results that are success or failure. If the number of trials m = 1, then it is called Bernoulli distribution that is special case for the Binomial distribution. Binomial distribution determines the probability of successful outcomes. It has two parameters, m and p where m denotes the number of trial and p denotes the success outcomes.