INTEGRATION OF THE NORMAL DISTRIBUTION CURVE By Tom Irvie Email: tomirvie@aol.com March 3, 999 Itroductio May processes have a ormal probability distributio. Broadbad radom vibratio is a example. The purpose of this report is to derive a formula for itegratig the ormal distributio curve. This effort is eeded due to the limitatios of statistical tables published i textbooks. Derivatio From Referece, the probability desity fuctio ( x;µ,σ) give by of a ormal distributio is x µ σ ( x; µ, σ), < x < σ π () here x is a cotiuous radom variable, µ is the mea, σ is the stadard deviatio. The ormal distributio curve has the shape sho i Figure. For this distributio, the probability P that the radom variable X has a value betee x x ad x x is obtaied by itegratig the area uder the probability desity curve. x < σ () σ π x ( x X < x ) ( x; µ, )dx P x x µ < x (3) σ π σ ( x X < x ) dx P No trasform the data via a radom variable Z.
µ Z X σ (4) dz dx σ (5) By substitutio, σ + µ σ σ + µ (6a) π (( + µ ) < ( σ Z + µ ) < ( σ + µ ) d P No assume a ero mea ad a uity stadard deviatio. < (6b) π ( Z < ) d P Note that the ormal distributio is symmetric about the mea. No cosider the case here > >. Further reuire that each limit be euidistat from the ero mea. Thus, -. Euatio (6c) ca be restated as (7) π ( < Z < ) d P Let (8) d d (9) () π ( < Z < ) { }d P here There are a umber of euivalet methods for evaluatig the itegral i euatio (). Oe method is by use of the error fuctio. The method used here is based o the icomplete gamma fuctio P(a,), as give i Referece.
P ( a, ) ( a, ) ( a) ( a) γ [ ( )] ( a t t ) [ ] dt, a > () Note that the gamma fuctio itself is a stadard mathematical fuctio defied as ( ) x x [ t ] ( t) dt, x > () a γ ( a,) [ t ] ( t) dt (3) No let u t (4) udu dt (5) By substitutio, (a ) (, ) [ u] [ ( u )][ u ] γ a du (6) (a ) (, ) [ ( u )][ u ] γ a du (7) No let a/. γ, ( ) u du (8) From Referece, the γ ( a,) fuctio ca be represeted i series form as a [ ][ ] ( a) (,) ( ) γ a ( ) (9) a + + γ, [ ( ) ][ ] 3 + () 3
4 Euatig the respective right-had-sides of euatio (8) ad (). ( ) ( ) [ ][ ] + 3 du u () Divide each side by. π ( ) ( ) [ ][ ] + π π 3 du u () No let y (3) ( ) ( ) [ ][ ] + π π y y 3 y y du u (4) Recall euatio () restated here as euatio (5). ( ) { }d Z P π < < (5) The right side of euatio (5) matches the left side of euatio (4) for u ad y. ( ) ( ) [ ][ ], 3 Z P + π < < (6) Thus, the probability of the radom variable Z ca be calculated via (6).
Note that π (7) Furthermore, the gamma fuctio follos a recursive relatioship, as give i Referece. ( + ) ( ) (8) Euatios (7) ad (8) allo for a simple computer program implemetatio of euatio (6). Tables of probability values based o this euatio are give i Appedix A. Examples. A certai maufacturig process yields a reliability of 6σ ith a ero mea. Ho may defects are ected per oe millio parts? The Table i Appedix A shos that the probability of a part fallig outside of the absolute value of 6σ is.973758e-9. Thus, the ected umber of part failures is., hich is essetially ero.. A.5σ shift i the mea occurs i the previous maufacturig process. Ho may defects are ected per oe millio parts cosiderig this shift? The limits are thus -7.5σ to +4.5σ. Note that the values could be iterchaged due to symmetry. For practical purposes, there is ero probability of a occurrece less tha -7.5σ. The Table i Appedix A shos that the probability of a part fallig outside of the absolute value of 4.5σ is 6.7953465E-6. The probability from +4.5σ to ifiity is oe-half this amout, or 3.3976735E-6. The umber of ected defects i a lot of oe millio is thus 3.4, rouded to to sigificat digits. Refereces. R. Walpole ad R. Myers, Probability ad Statistics for Egieers ad Scietists, d editio, Macmilla, Ne York, 978.. William Press, et al, Numerical Recipes The Art of Scietific Computig, Cambridge Uiversity Press, Cambridge, UK, 986. 5
Normal Distributio Curve Area uder the curve (x; µ, σ) -σ µ σ Figure. x 6
.. 7.978763E-3 9.987E-..5956674E- 9.8443373E-.3.3939468E- 9.766753E-.4 3.968737E- 9.68936E-.5 3.987767E- 9.6388E-.6 4.78443653E- 9.555635E-.7 5.586344E- 9.449366E-.8 6.376744E- 9.363756E-.9 7.7785E- 9.8875E-. 7.96556746E- 9.34435E-. 8.75965E- 9.49375E-. 9.55685E- 9.448348E-.3.3433573E- 8.9656647E-.4.34E- 8.8865999E-.5.935385E- 8.876465E-.6.7896E- 8.78874E-.7.34989863E- 8.6537E-.8.484743E- 8.575568E-.9.569869E- 8.49393E-..585949E- 8.44858E-..663337E- 8.33667673E-..748845E- 8.58755E-.3.8983E- 8.8977E-.4.89669743E- 8.3357E-.5.97465E- 8.587349E-.6.5366E- 7.94863774E-.7.839746E- 7.87654E-.8.5495E- 7.7947755E-.9.88376E- 7.78638E-.3.358844E- 7.647756E-.3.4343944E- 7.5656956E-.3.53669E- 7.4896833E-.33.58638E- 7.439996E-.34.664347E- 7.3385658E-.35.73663E- 7.6338698E-.36.85866E- 7.884734E-.37.88675E- 7.3849E-.38.9654585E- 7.394545E-.39 3.3463453E- 6.96536547E-.4 3.843483E- 6.895657E-.4 3.8945E- 6.885948E-.4 3.554546E- 6.74485454E-.43 3.384359E- 6.679564E-.44 3.46893E- 6.599377E-.45 3.4789559E- 6.5744E-.46 3.54483779E- 6.4556E- λ P[-λ < Z < λ] P[ Z > λ].47 3.664498E- 6.383558E-.48 3.687767E- 6.37393E-.49 3.75866E- 6.433899E-.5 3.89493E- 6.77577E-.5 3.89948538E- 6.546E-.5 3.9693645E- 6.363575E-.53 4.388869E- 5.9693E-.54 4.8968E- 5.89973E-.55 4.76866E- 5.839374E-.56 4.4556E- 5.75479438E-.57 4.333E- 5.68677698E-.58 4.388538E- 5.69468E-.59 4.448935E- 5.55965E-.6 4.5493764E- 5.485636E-.6 4.58389E- 5.48688E-.6 4.64743E- 5.3557787E-.63 4.73546E- 5.8694584E-.64 4.77874E- 5.7599E-.65 4.8437778E- 5.569E-.66 4.97467E- 5.95389E-.67 4.974E- 5.85779E-.68 5.3495539E- 4.965446E-.69 5.98583E- 4.99487E-.7 5.67696E- 4.839734E-.7 5.95864E- 4.777436E-.7 5.84754E- 4.754996E-.73 5.346985E- 4.653985E-.74 5.476E- 4.5999994E-.75 5.4674595E- 4.535475E-.76 5.574545E- 4.4754585E-.77 5.5877E- 4.499893E-.78 5.64695E- 4.3539875E-.79 5.7473E- 4.957768E-.8 5.76893E- 4.37797E-.8 5.85984E- 4.79476E-.8 5.87783893E- 4.67E-.83 5.93466E- 4.6538784E-.84 5.99963E- 4.98387E-.85 6.467494E- 3.953586E-.86 6.957E- 3.8978943E-.87 6.5699596E- 3.84344E-.88 6.469E- 3.788593E-.89 6.65344E- 3.73465886E-.9 6.3879749E- 3.685E-.9 6.377749E- 3.685E-.9 6.4474E- 3.5757759E-.93 6.476896E- 3.53784E- 7
.94 6.578439E- 3.47756E-.95 6.57887747E- 3.453E-.96 6.6944785E- 3.37555E-.97 6.6795358E- 3.34649E-.98 6.79388E- 3.7869E-.99 6.778588E- 3.749E-. 6.868949E- 3.7358E-. 6.87547E- 3.4959E-. 6.97539E- 3.77846E-.3 6.96989994E- 3.36E-.4 7.6699E-.983399E-.5 7.68887E-.93783E-.6 7.8554E-.8944599E-.7 7.53869E-.846939E-.8 7.98578E-.848E-.9 7.486856E-.757344E-. 7.8667878E-.733E-. 7.33974E-.669996E-. 7.378638E-.67376E-.3 7.453776E-.584764E-.4 7.4573699E-.54863E-.5 7.498569E-.54387E-.6 7.539594E-.464886E-.7 7.579993E-.4969E-.8 7.6999785E-.385E-.9 7.6595368E-.344639E-. 7.698666E-.33934E-. 7.7377E-.678893E-. 7.775355E-.464875E-.3 7.83895E-.86975E-.4 7.85466E-.4975394E-.5 7.887453E-.99547E-.6 7.933638E-.766936E-.7 7.959537E-.48463E-.8 7.99454864E-.54536E-.9 8.94934E-.975658E-.3 8.63993E-.936969E-.3 8.98464E-.995836E-.3 8.36498E-.868358E-.33 8.64879E-.83587E-.34 8.9754655E-.845345E-.35 8.9847E-.775983E-.36 8.6776E-.738994E-.37 8.93398E-.76869E-.38 8.343355E-.67586645E-.39 8.35473E-.6458877E-.4 8.3848668E-.65338E-.4 8.44637E-.58539683E- λ P[-λ < Z < λ] P[ Z > λ].4 8.443939E-.556768E-.43 8.47898E-.5779E-.44 8.536E-.49867399E-.45 8.59448E-.475859E-.46 8.557996E-.44974E-.47 8.5843846E-.456754E-.48 8.66753E-.3887347E-.49 8.63775764E-.36436E-.5 8.66385597E-.336443E-.5 8.68956576E-.34344E-.5 8.74894E-.85976E-.53 8.739837E-.6679E-.54 8.76439647E-.356353E-.55 8.78858484E-.456E-.56 8.849E-.875988E-.57 8.83584889E-.645E-.58 8.8589334E-.46866E-.59 8.886595E-.83485E-.6 8.9447E-.9598583E-.6 8.9644E-.7397856E-.6 8.9476773E-.5377E-.63 8.9689853E-.3497E-.64 8.98994833E-.567E-.65 9.5764E- 9.894936E-.66 9.385547E- 9.694455E-.67 9.58636E- 9.4993636E-.68 9.74684E- 9.957357E-.69 9.89745E- 9.79546E-.7 9.86974E- 8.93955E-.7 9.7347E- 8.765873E-.7 9.4567558E- 8.543446E-.73 9.636975E- 8.36375E-.74 9.8498E- 8.8598E-.75 9.988686E- 8.8337E-.76 9.5993E- 7.847866E-.77 9.37859E- 7.67747E-.78 9.49439E- 7.575967E-.79 9.654689E- 7.345394E-.8 9.83936E- 7.86638E-.8 9.9743E- 7.95787E-.8 9.34995E- 6.875949E-.83 9.3756E- 6.7499388E-.84 9.343763E- 6.5768373E-.85 9.3568645E- 6.4335496E-.86 9.374474E- 6.88556E-.87 9.38568E- 6.483879E-.88 9.39899E- 6.8779E-.89 9.444E- 5.875796E- 8
.9 9.456688E- 5.743396E-.9 9.43866787E- 5.63333E-.9 9.454E- 5.48578994E-.93 9.463936E- 5.3668378E-.94 9.4763E- 5.3796899E-.95 9.488388E- 5.769E-.96 9.54E- 4.9995793E-.97 9.5669E- 4.8838376E-.98 9.59647E- 4.773587E-.99 9.534964E- 4.6599355E-. 9.54499736E- 4.55639E-. 9.555688E- 4.443889E-. 9.56666E- 4.33833875E-.3 9.5764346E- 4.3565393E-.4 9.58649674E- 4.35357E-.5 9.59635569E- 4.364438E-.6 9.66459E- 3.9398548E-.7 9.6547656E- 3.8453445E-.8 9.6474467E- 3.755539E-.9 9.6338E- 3.6677997E-. 9.64759E- 3.57884E-. 9.654644E- 3.48583559E-. 9.65993955E- 3.46453E-.3 9.6688387E- 3.37634E-.4 9.6764533E- 3.3547667E-.5 9.68444785E- 3.55548E-.6 9.69733E- 3.776696E-.7 9.6999354E- 3.68459E-.8 9.774538E-.957466E-.9 9.7475763E-.854368E-. 9.7935E-.786895E-. 9.7894838E-.7563E-. 9.73583E-.6487676E-.3 9.745557E-.5747449E-.4 9.749977E-.5999E-.5 9.755555E-.44489453E-.6 9.7678749E-.3858E-.7 9.767947E-.37583E-.8 9.77393E-.676885E-.9 9.77978683E-.366E-.3 9.785578E-.448E-.3 9.79846E-.88854E-.3 9.796593E-.348773E-.33 9.893849E-.9865E-.34 9.8766E-.9837399E-.35 9.86589E-.87734E-.36 9.87565E-.874935E-.37 9.895E-.7788853E- λ P[-λ < Z < λ] P[ Z > λ].38 9.868736E-.73638E-.39 9.83567E-.6848378E-.4 9.836498E-.639578E-.4 9.8447479E-.59555E-.4 9.84479493E-.5557E-.43 9.84977E-.59888E-.44 9.853738E-.468769E-.45 9.8574379E-.48565E-.46 9.86698E-.389376E-.47 9.86488695E-.3535E-.48 9.868676E-.338383E-.49 9.87569E-.774395E-.5 9.8758669E-.49337E-.5 9.8796884E-.736E-.5 9.886457E-.7354834E-.53 9.88593747E-.4657E-.54 9.8894753E-.85469E-.55 9.89778E-.7799E-.56 9.8953784E-.46763E-.57 9.898349E-.69855E-.58 9.99968E- 9.8835E-3.59 9.9447E- 9.5975939E-3.6 9.967764E- 9.33765E-3.6 9.9945778E- 9.547E-3.6 9.973E- 8.799767E-3.63 9.94653E- 8.5384868E-3.64 9.979397E- 8.967E-3.65 9.99583E- 8.49779E-3.66 9.985935E- 7.84655E-3.67 9.944875E- 7.58547E-3.68 9.9637784E- 7.366E-3.69 9.9854798E- 7.459E-3.7 9.93665E- 6.9339476E-3.7 9.937679E- 6.7838E-3.7 9.934788E- 6.58963E-3.73 9.93666567E- 6.3334355E-3.74 9.938568E- 6.439844E-3.75 9.944474E- 5.9595647E-3.76 9.949864E- 5.78365E-3.77 9.9439437E- 5.65697E-3.78 9.94564E- 5.43588985E-3.79 9.947996E- 5.7846E-3.8 9.94889739E- 5.666E-3.8 9.954585E- 4.9545E-3.8 9.9597635E- 4.836495E-3.83 9.95345E- 4.65484E-3.84 9.95488647E- 4.535338E-3.85 9.956877E- 4.3799E-3 9
.86 9.9576359E- 4.3648E-3.87 9.958958E- 4.47799E-3.88 9.96348E- 3.976757E-3.89 9.964758E- 3.85486E-3.9 9.9668373E- 3.73666E-3.9 9.963857E- 3.648756E-3.9 9.96499686E- 3.53386E-3.93 9.96638E- 3.38964E-3.94 9.9677878E- 3.847E-3.95 9.9686E- 3.777399E-3.96 9.96936E- 3.76394E-3.97 9.973E-.97799749E-3.98 9.97756E-.8848383E-3.99 9.976E-.78977447E-3 3. 9.9734E-.6997966E-3 3. 9.9738753E-.64769E-3 3. 9.974753E-.5774686E-3 3.3 9.97554463E-.44553739E-3 3.4 9.976349E-.3657849E-3 3.5 9.977586E-.884366E-3 3.6 9.9778663E-.33699E-3 3.7 9.978594E-.45877E-3 3.8 9.9799994E-.7595E-3 3.9 9.97998435E-.56495E-3 3. 9.9864794E-.935643E-3 3. 9.9897E-.8787344E-3 3. 9.98949E-.8854E-3 3.3 9.985937E-.748633E-3 3.4 9.9835E-.68947835E-3 3.5 9.9836795E-.637463E-3 3.6 9.98439E-.5776939E-3 3.7 9.984756E-.5438938E-3 3.8 9.985749E-.47755E-3 3.9 9.985777E-.47794E-3 3. 9.986574E-.3747588E-3 3. 9.986765E-.373497E-3 3. 9.987894E-.89597E-3 3.3 9.987698E-.3798E-3 3.4 9.988473E-.9597E-3 3.5 9.9884595E-.5458E-3 3.6 9.98885878E-.44E-3 3.7 9.989455E-.7547484E-3 3.8 9.989699E-.38787E-3 3.9 9.989986E-.87383E-3 3.3 9.99335E- 9.6684885E-4 3.3 9.99674E- 9.39597E-4 3.3 9.999986E- 9.7448E-4 3.33 9.99354E- 8.6845984E-4 λ P[-λ < Z < λ] P[ Z > λ] 3.34 9.9966E- 8.37783899E-4 3.35 9.999884E- 8.8564E-4 3.36 9.99575E- 7.794475E-4 3.37 9.994838E- 7.568837E-4 3.38 9.99754E- 7.485898E-4 3.39 9.99374E- 6.989637E-4 3.4 9.99364E- 6.7385853E-4 3.4 9.993537E- 6.4968795E-4 3.4 9.99373789E- 6.6357E-4 3.43 9.9939649E- 6.35849E-4 3.44 9.994886E- 5.87487E-4 3.45 9.9943943E- 5.6586554E-4 3.46 9.9945985E- 5.475388E-4 3.47 9.9947954E- 5.458365E-4 3.48 9.99498586E- 5.43783E-4 3.49 9.9956979E- 4.83547E-4 3.5 9.9953474E- 4.655858E-4 3.5 9.9955893E- 4.486694E-4 3.5 9.99568453E- 4.3546799E-4 3.53 9.9958444E- 4.5559667E-4 3.54 9.99599873E- 4.73E-4 3.55 9.9964769E- 3.8535E-4 3.56 9.996945E- 3.7854794E-4 3.57 9.996439E- 3.56988E-4 3.58 9.9965646E- 3.435947E-4 3.59 9.996693E- 3.367796E-4 3.6 9.9968783E- 3.878E-4 3.6 9.9969383E- 3.6975E-4 3.6 9.9975397E-.94636E-4 3.63 9.9976579E-.834E-4 3.64 9.997736E-.76384E-4 3.65 9.9973776E-.6439E-4 3.66 9.99747785E-.5548E-4 3.67 9.9975745E-.455469E-4 3.68 9.99766766E-.3333954E-4 3.69 9.99775746E-.4545E-4 3.7 9.997844E-.5599467E-4 3.7 9.997974E-.75947E-4 3.7 9.998777E-.99778E-4 3.73 9.99885E-.947977E-4 3.74 9.998598E-.8455E-4 3.75 9.998365E-.7683457E-4 3.76 9.998387E-.6993357E-4 3.77 9.9983675E-.6347547E-4 3.78 9.998437E-.5688359E-4 3.79 9.99849353E-.564785E-4 3.8 9.9985534E-.4469688E-4 3.8 9.998633E-.3896679E-4
3.8 9.99866548E-.3345674E-4 3.83 9.9987857E-.84359E-4 3.84 9.99876966E-.3343E-4 3.85 9.998888E-.8785E-4 3.86 9.9988663E-.33875E-4 3.87 9.998965E-.8835353E-4 3.88 9.99895544E-.4456465E-4 3.89 9.99899756E-.44E-4 3.9 9.999387E- 9.69688E-5 3.9 9.999774E- 9.96E-5 3.9 9.99945E- 8.85489686E-5 3.93 9.999554E- 8.4945866E-5 3.94 9.999858E- 8.4869E-5 3.95 9.999849E- 7.8593E-5 3.96 9.99955E- 7.49497634E-5 3.97 9.99987E- 7.87638E-5 3.98 9.999385E- 6.89568E-5 3.99 9.9993397E- 6.673953E-5 4. 9.99936658E- 6.3344837E-5 4. 9.999398E- 6.787479E-5 4. 9.99948E- 5.89844E-5 4.3 9.999443E- 5.5776859E-5 4.4 9.99946549E- 5.3454E-5 4.5 9.9994878E- 5.7639E-5 4.6 9.999597E- 4.97759E-5 4.7 9.9995987E- 4.73377E-5 4.8 9.99954964E- 4.53578E-5 4.9 9.99956863E- 4.337389E-5 4. 9.99958685E- 4.3538E-5 4. 9.9996434E- 3.956597E-5 4. 9.99963E- 3.7887399E-5 4.3 9.9996374E- 3.676334E-5 4.4 9.9996569E- 3.47358E-5 4.5 9.9996675E- 3.347575E-5 4.6 9.9996875E- 3.847594E-5 4.7 9.9996954E- 3.4599639E-5 4.8 9.9997849E-.95996E-5 4.9 9.99975E-.78954455E-5 4. 9.9997339E-.669498E-5 4. 9.99974463E-.5537688E-5 4. 9.9997557E-.44338E-5 4.3 9.9997663E-.33693E-5 4.4 9.99977648E-.359787E-5 4.5 9.9997863E-.37755E-5 4.6 9.99979557E-.44694E-5 4.7 9.9998453E-.9547967E-5 4.8 9.99983E-.8689333E-5 4.9 9.999833E-.786738E-5 λ P[-λ < Z < λ] P[ Z > λ] 4.3 9.99989E-.77989E-5 4.3 9.99983675E-.6354546E-5 4.3 9.99984397E-.569E-5 4.33 9.9998589E-.49934E-5 4.34 9.9998575E-.44876E-5 4.35 9.99986386E-.363753E-5 4.36 9.99986994E-.36444E-5 4.37 9.99987575E-.446537E-5 4.38 9.999883E-.867939E-5 4.39 9.99988665E-.33566E-5 4.4 9.9998975E-.85878E-5 4.4 9.99989663E-.33769E-5 4.4 9.99993E- 9.8793E-6 4.43 9.9999577E- 9.43388E-6 4.44 9.99994E- 8.99588778E-6 4.45 9.999943E- 8.587894E-6 4.46 9.999984E- 8.959659E-6 4.47 9.999978E- 7.89597E-6 4.48 9.9999536E- 7.464339E-6 4.49 9.9999878E- 7.3736E-6 4.5 9.999935E- 6.7953465E-6 4.5 9.9999357E- 6.487657E-6 4.5 9.9999386E- 6.839634E-6 4.53 9.99994E- 5.89836865E-6 4.54 9.99994375E- 5.6548E-6 4.55 9.99994635E- 5.3645956E-6 4.56 9.99994885E- 5.5368E-6 4.57 9.999953E- 4.87745E-6 4.58 9.9999535E- 4.6497593E-6 4.59 9.99995568E- 4.4346E-6 4.6 9.99995775E- 4.4994E-6 4.6 9.99995973E- 4.66897E-6 4.6 9.9999663E- 3.837444E-6 4.63 9.99996343E- 3.65665733E-6 4.64 9.9999656E- 3.484978E-6 4.65 9.9999668E- 3.39359E-6 4.66 9.99996838E- 3.69384E-6 4.67 9.99996988E- 3.99735E-6 4.68 9.999973E-.8687497E-6 4.69 9.9999768E-.73549E-6 4.7 9.99997398E-.6649E-6 4.7 9.9999753E-.477679E-6 4.7 9.9999764E-.35844643E-6 4.73 9.99997755E-.459837E-6 4.74 9.99997863E-.3789E-6 4.75 9.99997966E-.346649E-6 4.76 9.9999864E-.9359959E-6 4.77 9.9999858E-.84599E-6
4.78 9.9999847E-.759595E-6 4.79 9.9999833E-.667833E-6 4.8 9.9999843E-.5866563E-6 4.8 9.9999849E-.59396E-6 4.8 9.99998564E-.43558E-6 4.83 9.99998635E-.365335E-6 4.84 9.999987E-.98399E-6 4.85 9.99998765E-.346474E-6 4.86 9.9999886E-.7385753E-6 4.87 9.99998884E-.59849E-6 4.88 9.99998939E-.68584E-6 4.89 9.9999899E-.835978E-6 4.9 9.999994E- 9.58366554E-7 4.9 9.9999989E- 9.76399E-7 4.9 9.9999935E- 8.65444E-7 4.93 9.9999978E- 8.9668E-7 4.94 9.999999E- 7.8578E-7 4.95 9.9999958E- 7.43485E-7 4.96 9.9999995E- 7.493797E-7 4.97 9.9999933E- 6.69597E-7 4.98 9.99999364E- 6.358473E-7 4.99 9.99999396E- 6.37996E-7 5. 9.9999947E- 5.733345E-7 5. 9.99999456E- 5.443354E-7 5. 9.99999483E- 5.674795E-7 5.3 9.999995E- 4.9479835E-7 5.4 9.99999534E- 4.6553847E-7 5.5 9.99999558E- 4.4865E-7 5.6 9.9999958E- 4.956477E-7 5.7 9.999996E- 3.97857E-7 5.8 9.9999963E- 3.77434868E-7 5.9 9.9999964E- 3.586356E-7 5. 9.9999966E- 3.3965348E-7 5. 9.99999678E- 3.5883E-7 5. 9.99999694E- 3.5535657E-7 5.3 9.999997E-.897477E-7 5.4 9.9999975E-.7473847E-7 5.5 9.9999974E-.6486459E-7 5.6 9.99999753E-.4694984E-7 5.7 9.99999766E-.3493994E-7 5.8 9.99999778E-.8859E-7 5.9 9.9999979E-.948E-7 5. 9.999998E-.998856E-7 5. 9.999998E-.888463E-7 5. 9.999998E-.78933E-7 5.3 9.9999983E-.69539E-7 5.4 9.99999839E-.6576597E-7 5.5 9.99999848E-.599E-7 λ P[-λ < Z < λ] P[ Z > λ] 5.6 9.99999856E-.44554E-7 5.7 9.99999864E-.3643757E-7 5.8 9.9999987E-.983888E-7 5.9 9.99999878E-.3636E-7 5.3 9.99999884E-.5868E-7 5.3 9.9999989E-.96536E-7 5.3 9.99999896E-.37675E-7 5.33 9.999999E- 9.87658E-8 5.34 9.9999997E- 9.94658E-8 5.35 9.999999E- 8.79543E-8 5.36 9.9999997E- 8.3957E-8 5.37 9.999999E- 7.8736643E-8 5.38 9.9999996E- 7.44858379E-8 5.39 9.9999993E- 7.4576834E-8 5.4 9.99999933E- 6.664897E-8 5.4 9.99999937E- 6.347493E-8 5.4 9.9999994E- 5.9599353E-8 5.43 9.99999944E- 5.6354533E-8 5.44 9.99999947E- 5.3857E-8 5.45 9.9999995E- 5.36986E-8 5.46 9.9999995E- 4.7634577E-8 5.47 9.99999955E- 4.535575E-8 5.48 9.99999957E- 4.535835E-8 5.49 9.9999996E- 4.933735E-8 5.5 9.9999996E- 3.79794E-8 5.5 9.99999964E- 3.5883373E-8 5.5 9.99999966E- 3.38999655E-8 5.53 9.99999968E- 3.379E-8 5.54 9.9999997E- 3.47654E-8 5.55 9.9999997E-.8566966E-8 5.56 9.99999973E-.69774658E-8 5.57 9.99999975E-.5473934E-8 5.58 9.99999976E-.458585E-8 5.59 9.99999977E-.7696E-8 5.6 9.99999979E-.43584E-8 5.6 9.9999998E-.3668E-8 5.6 9.9999998E-.995745E-8 5.63 9.9999998E-.8963E-8 5.64 9.99999983E-.756E-8 5.65 9.99999984E-.6447845E-8 5.66 9.99999985E-.537976E-8 5.67 9.99999986E-.479757E-8 5.68 9.99999987E-.34694739E-8 5.69 9.99999987E-.73935E-8 5.7 9.99999988E-.98744E-8 5.7 9.99999989E-.97664E-8 5.7 9.99999989E-.65456E-8 5.73 9.9999999E-.43655E-8
5.74 9.9999999E- 9.46765444E-9 5.75 9.9999999E- 8.943456E-9 5.76 9.9999999E- 8.439436E-9 5.77 9.9999999E- 7.97548E-9 5.78 9.99999993E- 7.476357E-9 5.79 9.99999993E- 7.3864433E-9 5.8 9.99999993E- 6.6349358E-9 5.8 9.99999994E- 6.478469E-9 5.8 9.99999994E- 5.884769E-9 5.83 9.99999994E- 5.547366E-9 5.84 9.99999995E- 5.8359E-9 5.85 9.99999995E- 4.95796E-9 5.86 9.99999995E- 4.6867344E-9 5.87 9.99999996E- 4.3579589E-9 5.88 9.99999996E- 4.66543E-9 5.89 9.99999996E- 3.8695464E-9 5.9 9.99999996E- 3.63546E-9 5.9 9.99999997E- 3.4773E-9 5.9 9.99999997E- 3.94684E-9 5.93 9.99999997E- 3.934777E-9 5.94 9.99999997E-.8595E-9 5.95 9.99999997E-.684553E-9 5.96 9.99999997E-.53799E-9 5.97 9.99999998E-.3753495E-9 5.98 9.99999998E-.337453E-9 5.99 9.99999998E-.98478E-9 6. 9.99999998E-.973758E-9 3