Krus, D.J. (2001) Is normal distribution due to Karl Gauss? Euler, his family of gamma functions, and place in history of statistics. Quality and Quantity: International Journal of Methodology, 35, 445-446.
Is normal distribution
due to Karl Gauss?
Euler, his family of gamma functions,
and place in history of statistics
David J. Krus
Abstract.- While Gauss (1777-1855) has primacy in explicating properties of the normal distribution, it is Euler (1707-1783) who is predominant with respect to analytical formulation of this function. This is discussed within the context of uniform rendering of computer algorithms for higher transcendental functions.
Rapid development of Microsoft compilers provides motivation for translating routines written in other languages to the Microsoft-supported code. To do that, one has to analyze relevant algorithms in far more detail than a casual reader of statistical literature. When rewriting algorithms for higher transcendental functions into a uniform code, their commonalities become quite apparent. After completing translation of several algorithms, I started to write code for the t-distribution. During this task I could not stop asking myself: ‘But where is the e?’
The theory of higher transcendental functions is due to Euler, with beta, gamma, and incomplete gamma functions at the center of this theory. The gamma functions have a general form
This conceptualization provides a common framework for a large number of functions, defined by assigning different values to the a, b, c, and d parameters, virtually ad libitum. One of the distributions within the family of the gamma functions is the t-distribution, normally written as
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In the above equation, the constant a is defined as
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and for degrees
of freedom approaching infinity, the limit of this constant is where
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The expression on the right side of the constant a can be rearranged as to accentuate the implied
series
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so we can see where the e is hiding. Since the limit of the exponent in the oblique brackets is -.5, the above equation can be written as
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Inserting the pro forma and replacing t with x, the above equation can
be written as Euler’s gamma function
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This equation describes a function identical to the function described by Gauss a generation later (in the h notation) as
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Historians can claim the predominance
of Gauss in explicating properties of the normal distribution and finding
applications for this function within astronomy, but hardly his primacy in
describing its general analytical form.
References
Euler, L. (1748) Introductio in analysin infinitorum.
Gauss, K. F. (1809) Theoria motus corporum coelestium.