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3 Distributions for X-Band Maritime Surveillance Radar Clutter
Published in Graham V. Weinberg, Radar Detection Theory of Sliding Window Processes, 2017
where t ≥ β > 0 and α > 0. This has been derived previously. The shape parameter governs the distribution’s shape, while the scale parameter determines where its support begins. Hence such a random variable has support [β,∞ $ [\beta , \infty $ ). By integration of (3.11), the corresponding distribution function is given by Fp(t)=1-(βt)α, $$ F_{p} (t) = 1 - (\frac{\beta }{t})^{\alpha } , $$
Monte Carlo Simulation
Published in Shyam S. Sablani, M. Shafiur Rahman, Ashim K. Datta, Arun S. Mujumdar, Handbook of Food and Bioprocess Modeling Techniques, 2006
Kevin Cronin, James P. Gleeson
Any probability distribution can be characterized by its parameters, which in turn can be estimated from the corresponding sample statistics. Generally, a probability distribution function has three parameters that can be geometrically interpreted as defining the location, scale, and shape of the distribution. A location parameter represents the position of the distribution on the x-axis by specifying an abscissa such as the minimum value or the average of the distribution. Changing the location parameter shifts the distribution left or right along the x-axis. The scale parameter represents the width or dispersion of the distribution (such as the standard deviation in the normal distribution). Changing this parameter compresses or expands the distribution without changing its basic shape. The shape parameter represents the shape of distribution usually characterized by the skewness of the distribution. Note that not all distributions have a shape parameter and some have more than one (such as the beta distribution).
Quality Control in Manufacturing
Published in Zainul Huda, Manufacturing, 2018
Gamma Distribution: The gamma distribution can be thought of as a waiting time between Poisson distributed events. It is a distribution that arises naturally in processes for which the waiting times between events are relevant. The gamma distribution can assume many different shapes depending on the values of the shape parameter (α) and the scale parameter (β). The mathematical expression for the failure density function for a gamma distribution is given elsewhere (Melchers, 1999). The mean of the gamma distribution (μG.d) is given by μG.d = α/β
Techno-socio-economic and sensitivity analysis of standalone micro-grid located in central India
Published in International Journal of Ambient Energy, 2023
Arun Rathore, Anupam Kumar, N. P. Patidar
For the representation of the wind speed curve, many density functions are employed. A popular distribution for describing wind speed is the two-parameter Weibull distribution function. The features of the wind speed data are well illuminated by the shape and scale parameters. As implied by its name, the shape parameter specifies the distribution’s shape. Use the scaling parameters to determine spread out of the distribution. Equation (6) represents Weibull PDF (Rathore and Patidar 2020, 2021). where, = Shape parameter of Weibull distribution; = Weibull probability density function for wind speed ; = Scale parameter of Weibull distribution; = Wind speed in m/s.
The shareability potential of ride-pooling under alternative spatial demand patterns
Published in Transportmetrica A: Transport Science, 2022
Jaime Soza-Parra, Rafał Kucharski, Oded Cats
Subsequently, potential destinations are generated around each centre. Their distance to the centre d follows a Gamma distribution with shape parameter kc and scale parameter sc, hence its probability density function is as follows: Where Γ(kc) corresponds to the Gamma function evaluated at the shape parameter kc: The scale parameter is fixed to represent the city under consideration and the shape parameter reflects the density. Gamma probability density function is selected as it properly describes the distance from a bivariate normal distributed coordinate to its centre. In addition, Gamma functions serve as good friction parameters in trip distribution models (e.g. Schiffer 2012) and have also been widely used in terms of travel time modelling (Guenthner and Hamat 1988; Kim and Mahmassani 2015). When the shape parameter is equal to one, it corresponds to an exponential distribution where the mean equals the scale of the distribution, and when it is larger than one, the distribution is skewed and varies with shape. Consequently, smaller shape parameter values represent the more concentrated destinations around each of the centres. Four specific distributions are illustrated in Figure 2a.
A synthetic unit hydrograph model based on fractal characteristics of watersheds
Published in International Journal of River Basin Management, 2019
I Gede Tunas, Nadjadji Anwar, Umboro Lasminto
represents the Gamma’s function, k is the shape parameter and θ is the scale parameter. According to probability theory and statistics, shape and scale parameters are special kinds of numerical parameters of a parametric family of probability distributions. Shape parameter plays an important role to control the general shape of the probability distribution, especially the slope and peak of the curve. Shape parameters allow a distribution to take on a variety of shapes, depending on the value of the shape parameter. Scale parameter determines the statistical dispersion of the probability distribution. If the scale parameter is large, then the distribution will be more spread out and if the scale parameter is small then it will be more concentrated (Kottegoda and Rosso 2008). The combination of the two parameters generates a distribution curve. For the constant k, the greater the value of the θ is, the more the abscissa of the curve spread out and vice versa. For the constant θ, the greater the value of the k is, the more the peak of the curve decreases and becomes blunt, and vice versa. This indicates that in their roles the two parameters have the same tendency in determining the shape of the distribution curve (Figure 4).