Accurate Approximation of Standard Normal Distribution's CDF, Study Guides, Projects, Research of Physics

Download Accurate Approximation of Standard Normal Distribution's CDF and more Study Guides, Projects, Research Physics in PDF only on Docsity! See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/324859407 Accurate Evaluation of the Cumulative Distribution of the Standard Normal Distribution. Article · May 2018 CITATIONS 0 1 author: Some of the authors of this publication are also working on these related projects: NEURON MODELS View project ANDRICA CONJECTURE View project Alvaro H. Salas National University of Colombia 190 PUBLICATIONS 938 CITATIONS SEE PROFILE All content following this page was uploaded by Alvaro H. Salas on 01 May 2018. The user has requested enhancement of the downloaded file. 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 on[-3, 3] Φ(x) = def 1 2 π  -∞ x ⅇ- t 2 2 ⅆ t ≈ x 2 - 25 867 x 4398 + 76 885 2911 x 2 - 10 219 x 2668 + 201 671 17 394 x 2 + 25 811 x 3468 + 243 814 12 965 x 2 + 25 453 x 3241 + 157 319 9299 x 2 + 23 209 x 2875 + 38 477 2340  2 x2 + 77 961 4811 x 2 - 5927 x 1480 + 97 966 5217 x 2 - 5151 x 2741 + 13 529 808 x 2 + 5151 x 2741 + 13 529 808 x 2 + 5927 x 1480 + 97 966 5217 for x ≤ 3 with an error less that 10-7. 4 Normal Distribution-CDF.nb In[102]:= Join{{x, Φ[x], Aproximant, Error}}, Tablex, SetPrecision 1 2 π  -∞ x ⅇ - t 2 2 ⅆt, 15, SetPrecision x2 - 25 867 x 4398 + 76 885 2911 x 2 - 10 219 x 2668 + 201 671 17 394 x 2 + 25 811 x 3468 + 243 814 12 965 x 2 + 25 453 x 3241 + 157 319 9299 x 2 + 23 209 x 2875 + 38 477 2340  2 x2 + 77 961 4811 x 2 - 5927 x 1480 + 97 966 5217 x 2 - 5151 x 2741 + 13 529 808 x 2 + 5151 x 2741 + 13 529 808 x 2 + 5927 x 1480 + 97 966 5217 , 9, 1 2 π  -∞ x ⅇ - t2 2 ⅆt - x2 - 25 867 x 4398 + 76 885 2911 x 2 - 10 219 x 2668 + 201 671 17 394 x 2 + 25 811 x 3468 + 243 814 12 965 x 2 + 25 453 x 3241 + 157 319 9299 x 2 + 23 209 x 2875 + 38 477 2340  2 x2 + 77 961 4811 x 2 - 5927 x 1480 + 97 966 5217 x 2 - 5151 x 2741 + 13 529 808 x 2 + 5151 x 2741 + 13 529 808 x 2 + 5927 x 1480 + 97 966 5217  // ReleaseHold, {x, 0, 3, 0.25} // TableForm Out[102]//TableForm= x Φ[x] Aproximant Error 0. 0.500000000000000 0.499999998 1.86274×10-9 0.25 0.598706325682924 0.598706323 2.82803×10-9 0.5 0.691462461274013 0.691462458 3.73845×10-9 0.75 0.773372647623132 0.773372643 4.46441×10-9 1. 0.841344746068543 0.841344741 4.91155×10-9 1.25 0.894350226333145 0.894350221 5.03919×10-9 1.5 0.933192798731142 0.933192794 4.85837×10-9 1.75 0.959940843136183 0.959940839 4.3617×10-9 2. 0.977249868051821 0.977249865 3.07476×10-9 2.25 0.987775527344955 0.987775529 -1.73651×10-9 2.5 0.993790334674224 0.993790353 -1.8543×10-8 2.75 0.997020236764945 0.997020295 -5.77451×10-8 3. 0.998650101968370 0.998650184 -8.16684×10-8 Normal Distribution-CDF.nb 5 View publication stats