OVERVIEW

The density function of the Gaussian, also called normal, distribution is given by

where m and ? are the mean and standard deviation of X, respectively, and satisfy the conditions ? ? < m < ? and ? > 0. The corresponding distribution function is given by

The distribution function can be determined in terms of the error function as

where

Letting u = ( x ? m)/ ? in (A.1), then

where

is the standard normal distribution with mean m = 0 and variance ? ² = 1, and also denoted (0,1). Tabulated values of I(x) and erf( x) are given in Tables A.1 ^[1] and A.2 ^[2], respectively.

Table A.1: Values of the Standard Normal Distribution Function

x	0	1	2	3	4	5	6	7	8	9
?3.0	0.0013	0.0010	0.0007	0.0005	0.0003	0.0002	0.0002	0.0001	0.0001	0.0000
?2.9	0.0019	0.0018	0.0017	0.0017	0.0016	0.0016	0.0015	0.0015	0.0014	0.0014
?2.8	0.0026	0.0025	0.0024	0.0023	0.0023	0.0022	0.0021	0.0021	0.0020	0.0019
?2.7	0.0035	0.0034	0.0033	0.0032	0.0031	0.0030	0.0029	0.0028	0.0027	0.0026
?2.6	0.0047	0.0045	0.0044	0.0043	0.0041	0.0040	0.0039	0.0038	0.0037	0.0036
?2.5	0.0062	0.0060	0.0059	0.0057	0.0055	0.0054	0.0052	0.0051	0.0049	0.0048
?2.4	0.0082	0.0080	0.0078	0.0075	0.0073	0.0071	0.0069	0.0068	0.0066	0.0064
?2.3	0.0107	0.0104	0.0102	0.0099	0.0096	0.0094	0.0091	0.0089	0.0087	0.0084
?2.2	0.0139	0.0136	0.0132	0.0129