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This Demonstration shows the probability density function (PDF) and the complementary cumulative distribution function (CCDF) of the ... distribution, given by ... ... and ... , respectively, with ... and ... as defined in the Details section. These functions can be viewed as a generalization of the ordinary exponential, which is recovered in the limit as ... , and present a power-law tail as ... . The exponent ... quantifies the curvature (shape) of the distribution, which is less pronounced for lower values of the parameter, and more pronounced for higher values. The constant ... ... is a characteristic scale, since its value determines the scale of the probability distribution: if ... is small, then the distribution will be more concentrated around the mode; if ... is large, then it will be more spread out. Finally, the parameter ... measures the heaviness of the right tail: the larger its magnitude, the fatter the tail. Assuming ... , the Demonstration shows the effects of different values of the parameters ... and ... on the shape of the PDF (on linear scale) and CCDF (both on a linear and log-log scale). In the last few years the ... distribution has appeared in a diverse range of applications, including both physical and systems and those in other fields.
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