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Accurate Evaluation of the Cumulative
Distribution of the Standard Normal
Distribution.
Article · May 2018
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Alvaro H. Salas
National University of Colombia
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Accurate Evaluation of the
Cumulative Distribution of the
Standard Normal Distribution.
By : Alvaro H. Salas
Universidad Nacional de Colombia.
https : //
en.wikipedia.org / wiki / Normal _ distribution
Approximation Formulas
First Approximation Formula
In[100]:= Join x, HoldForm ^1 2 π
- ∞
x ⅇ -^
t 2 (^2) ⅆ t , Aproximant, Error ,
Table x, SetPrecision ^1 2 π
x ⅇ -^
t 2 (^2) ⅆ t, 16 , SetPrecision
739 x 5 196 560 2 π
+^55 x^
4 8736
+^17 x^
3 468 2 π
+^95 x^
2 936
+ x 2 π
+^1
^55 x^
4 4368
+^95 x^
2 468
2 π
- ∞
x ⅇ -^
t 2 (^2) ⅆ t - 739 x^
5 196 560 2 π
+ 55 x^
4 8736
+ 17 x^
3 468 2 π
+ 95 x^
2 936
+ x 2 π
+^1
55 x 4 4368
+ 95 x^
2 468
+ 1 // ReleaseHold, { x, 0, 1, 0.05 } // TableForm
Out[100]//TableForm=
x ∫-∞
x (^) ⅇ - t (^2 2) ⅆ t 2 π Aproximant Error
- 0.5000000000000000 0.5000000000000000 0. 0.05 0.5199388058383725 0.5199388058383725 0. 0.1 0.5398278372770290 0.5398278372770291 - 2.22045 × 10 0.15 0.5596176923702425 0.5596176923702434 - 8.88178 × 10 0.2 0.5792597094391030 0.5792597094391225 - 1.95399 × 10 0.25 0.5987063256829237 0.5987063256831474 - 2.2371 × 10 - 0.3 0.6179114221889527 0.6179114221905960 - 1.64346 × 10 0.35 0.6368306511756191 0.6368306511844571 - 8.83804 × 10 0.4 0.6554217416103242 0.6554217416481328 - 3.78086 × 10 0.45 0.6736447797120800 0.6736447798478249 - 1.35745 × 10 0.5 0.6914624612740131 0.6914624616982843 - 4.24271 × 10 0.55 0.7088403132116536 0.7088403143965819 - 1.18493 × 10 0.6 0.7257468822499264 0.7257468852646274 - 3.0147 × 10 - 0.65 0.7421538891941353 0.7421538962844073 - 7.09027 × 10 0.7 0.7580363477769270 0.7580363633687422 - 1.55918 × 10 0.75 0.7733726476231318 0.7733726799728918 - 3.23498 × 10 0.8 0.7881446014166033 0.7881446652066814 - 6.37901 × 10 0.85 0.8023374568773076 0.8023375771413779 - 1.20264 × 10 0.9 0.8159398746532405 0.8159400925093920 - 2.17856 × 10 0.95 0.8289438736915182 0.8289442544580521 - 3.80767 × 10
- 0.8413447460685429 0.8413453904324962 - 6.44364 × 10
We see that 1 2 π
x ⅇ -^
t 2 (^2) ⅆ t <
739 x 5 196 560 2 π
+ 55 x^
4 8736
+ 17 x^
3 468 2 π
+ 95 x^
2 936
+ x 2 π
+^1
55 x^
4 4368
+ 95 x^
2 468
Second Approximation Formula
on[-3, 3]
Φ( x ) =^ def^
2 π
x ⅇ -^ t^22 ⅆ t ≈
x^2 -
25 867 x 4398
x^2 -
10 219 x 2668
x^2 +
25 811 x 3468
x^2 +
25 453 x 3241
x^2 +
23 209 x 2875
2 x^2 +
x^2 -
5927 x 1480
x^2 - 5151 x 2741
x^2 + 5151 x 2741
x^2 + 5927 x 1480
for x ≤ 3
with an error less that 10 -^7.
