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Statistical Data Analysis
Published in Timothy Bower, ®, 2023
The rand(n) function creates a n×n matrix of uniformly distributed random numbers in the interval (0 to 1). rand(m, n) creates a m×n matrix or vector. To generate n random numbers in the interval (a,b) use the formula r = a + (b-a)*rand(1, n). A uniform distribution means that each number is equally likely.
Estimation and inference
Published in Andrew Metcalfe, David Green, Tony Greenfield, Mahayaudin Mansor, Andrew Smith, Jonathan Tuke, Statistics in Engineering, 2019
Andrew Metcalfe, David Green, Tony Greenfield, Mahayaudin Mansor, Andrew Smith, Jonathan Tuke
Populations are modeled by probability distributions. A probability distribution is generally defined as a formula involving the variable x, say, and a few symbols which take specific values in an application. These symbols are known as parameters and in conjunction with the general formula, the parameters determine numerical characteristics of a population such as its mean, standard deviation, and quantiles, and in the case of binary data a proportion with a particular attribute. For example, a normal distribution has two parameters which are its mean μ and standard deviation σ. The exponential distribution for the time between events in a Poisson process is often defined by the rate parameter λ, events per unit time, in which case its mean is 1/λ time units. The uniform distribution is defined by two parameters a and b, which specify the range of the variable, and its mean is (a + b)/2.
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Published in Harald Paganetti, Proton Therapy Physics, 2018
A probability density function (PDF) expresses the relative likelihood that a variable will have a certain value, as determined by a random number. In mathematical terms, PDF(x) represents the probability of finding the random variable x′ within dx of a given value x. Considering a PDF(x) defined for a certain interval [a,b], the goal is to sample randomly between a and b according to the PDF(x) (a and b being finite and b > a). In case of a continuous uniform distribution, one obtains PDF(x) = 1/(b − a) for a ≤ x ≤ b and 0 for x < a and x > b. Assuming that random numbers Ri generated by a random number generator are uniformly distributed in [0,1], one can obtain random events via PDF(x) dx = dCDF. Thus, CDF(x)=∫axPDF(x′)dx′
Planning offline inspection strategies in low-volume manufacturing processes
Published in Quality Engineering, 2020
Elisa Verna, Gianfranco Genta, Maurizio Galetto, Fiorenzo Franceschini
Consider for example a case with only one output variable, denoted as Y, and one input variable, called X. The relationship between the two variables is given by the function Y = f(X). However, in realistic cases, this function is not exactly defined, that is, the coefficients of the mathematical model are affected by uncertainty. Furthermore, also the optimal value of the input variable (x*), that is, the value that optimizes the response output, is not exactly defined because of the uncertainty of the measurement device. For that reason, a variability range must be associated to it (by defining an upper UL and a lower LL variation limit, as illustrated in Figure 2). The probability distribution associated to X depends on the characteristics of the input variable. For instance, if the values are all equiprobable in the interval, a uniform distribution should be considered. As shown in Figure 2, the variance of the probability distribution of the output variable may be estimated by composing the uncertainties associated to both the input variable and the mathematical function, through the law of composition of variances (Ver Hoef 2012).
Reliability analysis of riprap stability around bridge piers
Published in Journal of Applied Water Engineering and Research, 2019
Mojtaba Karimaei Tabarestani, Amir Reza Zarrati
For this particular bridge site, it is believed that the flow angle may be as high as 30° (Johnson and Dock 1998 and Muzzamil et al. 2008). On the other hand, practically, it is not easy to estimate the flow angle due to irregularity in the river section and alignment upstream of the bridge site. Therefore, in order to examine the worst condition, a normal distribution with a Mean of 30 is considered for flow skew angle. The Variance is assumed equal to 0.1. Furthermore, it is assumed that the riprap stone density has a uniform distribution between 2500 (kg/m3) as lower and 3000 (kg/m3) as an upper limit. The uniform distribution (also called rectangular distribution) has a constant PDF between its two upper and lower limits.