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Phenomenological Creep Models of Fibrous Composites (Probabilistic Approach)
Published in Leo Razdolsky, Phenomenological Creep Models of Composites and Nanomaterials, 2019
A binomial experiment is one that possesses the following properties: the experiment consists of n repeated trials; each trial results in an outcome that may be classified as a success or a failure (hence the name, binomial); the probability of a success, denoted by p, remains constant from trial to trial and repeated trials are independent. The number of successes X in n trials of a binomial experiment is called a binomial random variable. In probability theory and statistics, the Bernoulli distribution, named after the Swiss scientist Jacob Bernoulli, is a discrete probability distribution, which takes value 1 with success probability p and value 0 with failure probability q = 1 - p. So if X is a random variable with this distribution, we have: Pr(X=1)=1-Pr(X=0)=1-q=p $$ Pr(X = 1) = 1 - Pr(X = 0) = 1 - q = p $$
Probability Theory
Published in A. C. Faul, A Concise Introduction to Machine Learning, 2019
The Bernoulli distribution describes an experiment where the outcome is binary. We can now consider the question of how many successes (x = 1) are in a set of N experiments. When considering m successes, there are N possibilities for the first success, N − 1 for the second, and so on until there are N − m + 1 possibilities for the last success. The resulting number N(N − 1) ⋯ (N − m + 1) is the number of m-permutationsm@m-permutation. It gives the number of possibilities of choosing an ordered set of size m from a set of size N. However, the ordering in our case is unimportant, and we therefore divide by the number of possible permutations of size m which is given by the factorialfactorialm! = m(m − 1) ⋯ 1. The resulting number of possibilities to achieve m times x = 1 in N experiments is given by the binomial coefficientbinomial coefficient, (Nm)=N(N−1)⋯(N−m+1)m(m−1)⋯1. It is read as “N choose m” and is the coefficient of the am term in the expansion of (1 + a)N.
Probability Distributions
Published in Alan R. Jones, Probability, Statistics and Other Frightening Stuff, 2018
The primary use for the Binomial Distribution is in modelling the number of successes or failures in a number of situations, such as test results. In particular, in the special case of a single trial or test, the Binomial Distribution is referred to as the Bernoulli Distribution. This is used extensively in modelling the probability of occurrence of risks and opportunities as inputs to Monte Carlo Modelling (see Volume V Chapter 3.)
A Bayesian approach to forward and inverse abstract argumentation problems
Published in Journal of Applied Non-Classical Logics, 2022
It is obvious that if and if . The distribution is called a Bernoulli distribution (Uspensky, 1937), often used to represent a discrete probability distribution with binary values. One can assume the uniform distribution, i.e. , when no knowledge on the presence of the attack relation in m is available or assumed. One can alternatively apply the result of textual analysis with natural language processing to give a value to each . We leave the values of unspecified to make our idea open to various application scenarios. Note that, as will be illustrated in Example 3.11, is updated in accordance with data. Collecting data is thus more important than trying to accurately specify the values of in our Bayesian approach.
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.