Transcription of The Pareto Distribution - American University
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The Pareto Distribution - American University
The Pareto FunctionConsider an arbitrary power function, x k x , where k is a constant and the exponent governs the relationship. The logarithmic transformation of this power function is linear in Log[x]. That is, if y= k x , then Log[y]= Log[k]+ Log[x]. Another way to say this is that the elasticity of y with respect to x is constant: d Log[y]d Log[x]= . To communicate the property that the elasticity does not depend on the size of x, the power relationship is called scale invariant. [ ]= TransformTwo Power Functions(square vs square root) LawA power law is a theoretical or empirical relationship governed by a power function. Examples include the following. In geometry, the area of a regular polygon is proportional to the square of the length of a side. In physics, the gravitational attraction of two objects is inversely proportional to the square of their distance. In ecology, Taylor s Law states that the variance of density is a power-function of mean population density.
The Pareto Distribution Background Power Function Consider an arbitrary power function, x↦kxα where k is a constant and the exponent α gov- erns the relationship. Note that if y=kxα, then Log[y]=Log[k]+αLog[x].That is, the logarith-
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