INTEGRATION OF THE NORMAL DISTRIBUTION CURVE

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1 INTEGRATION OF THE NORMAL DISTRIBUTION CURVE By Tom Irvie 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.

2 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.

3 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 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).

5 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 e-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 E-6. The probability from +4.5σ to ifiity is oe-half this amout, or E-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, William Press, et al, Numerical Recipes The Art of Scietific Computig, Cambridge Uiversity Press, Cambridge, UK,

6 Normal Distributio Curve Area uder the curve (x; µ, σ) -σ µ σ Figure. x 6

7 E E E E E E E E E E E E E E E E E E E E E E E E E E-.4.34E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E- λ P[-λ < Z < λ] P[ Z > λ] E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E- 4.67E E E E E E E E E E E E E E E E E E E E E E E- 7

8 E E E E E E E E E E E E E E E E E E E- 3.36E E E E E E E E E E-.848E E E E-.733E E E E E E E E E E E E E E-.4969E E-.385E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E- λ P[-λ < Z < λ] P[ Z > λ] E E E-.5779E E E E E E E E E E E E E E E E E E E E-.6679E E E E-.456E E E E-.645E E E E E E E E E E-.5377E E-.3497E E-.567E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E- 8

9 E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-.7563E E E E E E-.5999E E E E-.3858E E E E E E-.366E E-.448E E E E E E-.9865E E E E E E E E E- λ P[-λ < Z < λ] P[ Z > λ] E E E E E E E E E-.5557E E E E E E E E E E-.3535E E E E E E E E-.736E E E E-.4657E E E E-.7799E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-3 9

10 E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-.7595E E E E E E E E-.8854E E E E E E E E E E E E E E E E E E E E E E-.3798E E-.9597E E-.5458E E-.44E E E E E E E E E E E E E E E-4 λ P[-λ < Z < λ] P[ Z > λ] E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E- 4.73E E E E E E E E E E E E E E E E E E-.834E E E E-.6439E E-.5548E E E E E E-.4545E E E E E E E E E E-.8455E E E E E E E E E E E E E E E-4

11 E E E E E-.3343E E-.8785E E E E E E E E-.44E E E E- 9.96E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-5 λ P[-λ < Z < λ] P[ Z > λ] E E E E E-.569E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-.6649E E E E E E E E-.3789E E E E E E E-6

12 E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-.8859E E-.948E E E E E E E E E E E E-.599E-7 λ P[-λ < Z < λ] P[ Z > λ] E E E E E E E-.3636E E-.5868E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-.7696E E E E-.3668E E E E-.8963E E-.756E E E E E E E E E E E E E E E E E E E-8

13 E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-.8595E E E E E E E E E E E E E-9 3

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