Quality & Quantity 35: 445–446, 2001.
© 2001 Kluwer Academic Publishers. Printed in the Netherlands.
445
Research Note
Is Normal Distribution due to Karl Gauss? Euler,
his Family of Gamma Functions, and Place in
History of Statistics
DAVID J. KRUS
Arizona State University, U.S.A.
Abstract. While Gauss (1777–1855) has primacy in explicating properties of the normal distri-
bution, it is Euler (1707–1783) who is predominant with respect to analytical formulation of this
function. This case 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 statist-
ical literature does. When rewriting algorithms for higher transcendental functions
into a uniform code, their commonalities become quite apparent. After complet-
ing 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
φ(x) = ax
b
e
−cx
d
.
This conceptualization provides a common framework for a large number of func-
tions, 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
φ(t) =
1
√
νπ
ν+1
2
ν
2
1 +
t
2
ν
−(ν+1)/2
.
In the above equation, the constant a can be written as
a =
1
√
νπ
ν+1
2
ν
2
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