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Review of Probability Theory
Published in Harry G. Perros, An Introduction to IoT Analytics, 2021
The cumulative probability distribution of the geometric distribution isFXk=∑i=0kpi1−p=1−p∑i=0kpi=1−p1−pk+11−p=1−pk+1.
Some Probability Concepts for Engineers
Published in Richard L. Shell, Ernest L. Hall, Handbook of Industrial Automation, 2000
The geometric distribution arises when we are interested in the number of Bernoulli trials that are required until we get the first success. Now suppose that we define the random variable X as the number of Bernoulli trials that are required until we get the rth success. For the rth success to occur at the xth trial, we must have (r − 1) successes in the (x − 1) previous trials and one success in the rth trial (see Fig. 8). This random variable is called the negative binomial random variable and is denoted by NB(r, p). Its pmf is given in Table 2. Note that the gometric distribution is a special case of the negative binomial distribution obtained by setting (r = 1), that is, G(p) = NB(1, p).
Probability and Statistics
Published in José Guillermo Sánchez León, ® Beyond Mathematics, 2017
We use the geometric distribution (GeometricDistribution[p]) to model the number of trials until success, given a probability of success of p. The expected number of cars will be the mean of the GeometricDistribution with p = 0.1. It can be computed in two different ways: Mean[GeometricDistribution [0.1]] 9. Expectation[x, x ≈ GeometricDistribution[0.1]] 9.
Convergence analysis of max-consensus algorithm in probabilistic communication networks with Bernoulli dropouts
Published in International Journal of Systems Science, 2019
Amirhosein Golfar, Jafar Ghaisari
If is a Bernoulli random process with success probability , the number of trials on needed to get one success is the geometric distribution with parameter . If has a geometric distribution with parameter , the following results are concluded according to De Finetti (2017): where (1) is the expectation of geometric distribution and (2) is the cumulative distribution function of geometric distribution.
Determining the optimum process mean in a two-stage production system based on conforming run length sampling method
Published in Quality Technology & Quantitative Management, 2018
Mohammad Saber Fallah Nezhad, Zahra Davoodi Farsani
Consider a two-stage production system with serial production. In this system, first of the production process runs on the first machine and another part of the production process runs on a second machine. After processing by the first machine, assume that xk is defined as the number of conforming items between the successive (k – 1)-th and k-th defective items and the lot either rejected or passed over to the second machine. Also after processing by the second machine, assume that yk is defined as the number of conforming items between the successive (k – 1)-th and k-th defective items and the lot either rejected or accepted. Both parameters xk and yk follow the geometric distribution with probability of success p1 and p2, respectively. According to definition of geometric distribution in statistics and probability science, geometric distribution is based on the number of failures until the first success occurs. Then in here, the production of conforming items is considered as failure events and detecting the first defective item is considered as event of the first success therefore the number of conforming items until the detection of the first defective item follows geometric distribution. Thus the probability of success is equal to the defective Proportion pi (Proportion of the defective items in the lot at the stage i).