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Basics of probability and statistics
Published in Amit Kumar Gorai, Snehamoy Chatterjee, Optimization Techniques and their Applications to Mine Systems, 2023
Amit Kumar Gorai, Snehamoy Chatterjee
An important probability measure is the cumulative distribution function (CDF). The CDF of a discrete random variable, FX(k) represents the probability that the random variable X is less than or equal to k. It can be represented mathematically, as P(X≤k)=FX(k)=∑x=0kP(x)
Establishing threshold values for use in structural health monitoring
Published in Joan-Ramon Casas, Dan M. Frangopol, Jose Turmo, Bridge Safety, Maintenance, Management, Life-Cycle, Resilience and Sustainability, 2022
J. Chen, M.J. Chajes, H.W. Shenton
Where Fix is the CDF of the ith Gaussian distribution with mean μi and standard deviation σi. The inverse of the CDF (i.e., F-1) is the quantile function. For a given probability percentage value, the corresponding value of the x can be calculated using the quantile function. In this work, the threshold values are set as 99% USLs of the Gaussian mixture distribution models for the IRIB.
Process Excellence
Published in James William Martin, Operational Excellence, 2021
We will discuss two simple examples, a single operation and a workflow that consists of three sequential operations. In Figure 5.7, a single operation and independent variable, cycle time, has been assigned probability values over its observed range of cycle times. A specific cycle time of twelve days is used as an example. This simulation model is based on a uniform distribution and generates random numbers with a uniform occurrence having a probability between 0 and 1. These random numbers are transformed using the cumulative density function (cdf) of cycle times of the actual observed distribution. The cdf has a range between 0 and 1 based on the original probability density function (pdf). The relationship of a cdf to an independent variable can be discrete or continuous depending on the pdf, which is based on the distribution assumption. This example uses a uniform distribution. The functional relationship between cycle time and its occurrence probability has been discretely defined in Table 5.5. A random number in the range of greater than 0.539828 and less than 0.617911 is defined as a discrete cycle time of 12 days. Using a continuous “cdf,” we also map a one-to-one relationship between the continuous random variable in the range between 0 and 1 to a specific cycle time. This is shown in Figure 5.7.
Development and application of a modularized geometry optimizer for future supercritical CO2 turbomachinery optimization
Published in Engineering Applications of Computational Fluid Mechanics, 2022
Jianhui Qi, Jinliang Xu, Kuihua Han, Jingzhi Zhang
In probability theory and statistics, the cdf of a random variable X, evaluated at x, is the probability that X will take a value less or equal to x. In the case of a continuous distribution, it gives the area under the probability density function from minus infinity to x. The cumulative distribution function of a normal distribution is where μ shows the value where the cdf is 0.5, the mean of the corresponding standard deviation function. If we set that guarantees around 50% state vector points will be calculated through interpolation methods, while the rest 50% will be evaluated through high fidelity CFD simulations. σ is the standard deviation of the cdf. If we set , i.e. , the cdf is at 2.5% and 97.5% at and .
Uncertainty quantification: data assimilation, propagation and validation of the numerical model of the Arade river cable-stayed bridge
Published in Structure and Infrastructure Engineering, 2022
Iviane Cunha e Santos, José Luis Vital de Brito, Elsa de Sá Caetano
The cumulative distribution function (CDF) indicates the probability of a given event in the population. For this reason, the ordinate axis always ranges between 0 and 1. The CDF chart shown in Figure 16 describes the probability distribution of a real-valued random variable. The Box-Whiskers chart shows the distribution of data in an effective way, summarizing certain information, such as the mean and confidence interval, the quartiles and the outliers. The latter are the values that fall out of an interval centred in the mean and with semi-amplitude equal to 1.5 of the standard deviation. Figure 17 contains several graphic elements giving a good sense of the distribution of the set of data. The top and bottom lines of the ‘box’ are the lower and upper quartile values of the sample (i.e. 25th and 75th percentiles). Hence the distance between the two lines gives the interquartile range. The line in the middle of the box (50th percentile) is the median. The median cuts the distribution precisely in half, such that an equal number of points is more significant than this value as there are smaller than this value.
Semiparametric Models for Accelerated Destructive Degradation Test Data Analysis
Published in Technometrics, 2018
Yimeng Xie, Caleb B. King, Yili Hong, Qingyu Yang
We can also derive the failure time distribution from the semiparametric model. The event that the failure time T is less than t (i.e., T ⩽ t) is equivalent to that the degradation measurement at time t is less than the failure threshold (i.e., ), for monotonic decreasing degradation paths. Here, yt is the degradation measurement at t. Hence, the cumulative distribution function (cdf) of failure time, FT(t), can be calculated as where Φ( · ) is the cdf of the standard normal distribution. The quantile function can then be calculated as the inverse of the cdf. That is, the α quantile is tα = F− 1T(α). In the case of no closed-form expression, numerical methods can be used to find the quantile function from the cdf.