A METHOD OF APPROXIMATING PERCENTILE POINTS OF HIGHLY SKEWED, TWO PARAMETER BETA DISTRIBUTIONS. Thornton Boyd

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1 A METHOD OF APPROXIMATING PERCENTILE POINTS OF HIGHLY SKEWED, TWO PARAMETER BETA DISTRIBUTIONS by Thornton Boyd

United States Navai Postgraduate School T '

3 United States Navai Postgraduate School T ' :i r :'.r: A METHOD OF APPROXIMATING PERCENTILE POINTS OF HIGHLY SKEWED, TWO PARAMETER BETA DISTRIBUTIONS by Thornton Boyd Thesis Advisor W. M. Woods March 1971 Approved ^oh. public ^ididasc; dl&txlbiullon antimixad

, University of Texas, 1963 Submitted in partial fulfillment of the requirements for

5 A Method of Approximating Percentile Points of Highly Skewed, Two Parameter Beta Distr ibutions by Thornton ^Boyd Captain, United States Marine Corps B.A., University of Texas, 1963 Submitted in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE IN OPERATIONS RESEARCH from the NAVAL POSTGRADUATE SCHOOL March 1971

6 C"\

are known. A specific application is presented by utilizing the formula to compute the lower 100(1-0!

7 LIBRARY POSTGRADUATE SCHOOL MONTEREY, CALIF ABSTRACT An approximation formula is derived which provides a simplified and efficient method of calculating the O'th percentile point of a two parameter beta function when the two parameters are known. A specific application is presented by utilizing the formula to compute the lower 100(1-0!)% Bayesian confidence limit for the reliability of a component when the prior distribution of that reliability is known to be beta with parameters a' and b*. The posterior distribution is then determined by mission testing n items and recording the number of successes, s. This distribution is known to be beta with parameters a and b, where a = a' + s and b = b' + n - s. Therefore, the formula can be utilized to determine the Ofth percentile point of this posterior distribution which by definition is the lower 100(l-a)% confidence limit for the reliability of the component.

THE POSTERIOR DISTRIBUTION -8 III. THE APPROXIMATION FORMULA 10" A. DERIVATION 10 B.

9 TABLE OF CONTENTS I. INTRODUCTION 5 II. DISCUSSION - -7 A. BAYESIAN RELIABILITY 7 B. APPROXIMATING A BINOMIAL DISTRIBUTION WITH A BETA DISTRIBUTION-- 7 C. THE POSTERIOR DISTRIBUTION -8 III. THE APPROXIMATION FORMULA 10" A. DERIVATION 10 B. ACCURACY RESULTS 13 C. ALTERNATE APPROXIMATION FORMULAS 17 IV. EXAMPLES 20 A. EXAMPLE 1 20 B. EXAMPLE 2 20 BIBLIOGRAPHY INITIAL DISTRIBUTION LIST 23 FORM DD

11 LIST OF TABLES I. Comparison of Approximate and Exact 5th Percentile Points for the Beta Distribution l4 II. Comparison of Approximate and Exact 10th Percentile Points for the Beta Distribution 15 III. Comparison of Approximate and Ex^ct 20th Percentile Points for the Beta Distribution 16 IV. Comparison of Approximate and Exact 20th Percentile Points Generated by Selected Approximation Formulas for the Beta Distribution 19

distribution. More specif icallv, the prime interest was in approximating percentile points of highly skewed, two parameter beta distributions.

13 , +. I. INTRODUCTION The object of this paper was to determine a method of approximating the ( th percentile point for tha reliability of a component when this reliability is known to have a beta distribution. More specif icallv, the prime interest was in approximating percentile points of highly skewed, two parameter beta distributions. The approximation formula that was derived provides a simplified method for determining the CKth percentile point of a beta distribution when the two parameters a and b are known. Given that the reliability R of a component is distributed beta with parameters a and T b T the posterior distribution of R can then be determined. Let n = the number to be sampled Let s = the number of successes Let f = the number of failures The n items are then mission tested. By applying Bayes Theorem, the posterior distribution of R is found to be beta with parameters (a* + s) and (b' + f ) Let a = a f + s and b = b' + f Therefore, the posterior distribution of R is beta with parameters a and b. Once the values of a and b have been determined, the formula below can be utilized to approximate p, the #th percentile point of the reliability of the component. Arcs in M - I 2da+b-l a + b/' 3(a+b-l) 1

15 Where Z, is a number such that PjZ > * Z Z " = 1 -d. and Z is l-cj distributed normal (0,1). Therefore, Z can be looked up in any Standard Normal Tables.

(rjs), (s r), R ) ) II. DISCUSSION A. BAYESIAN RELIABILITY Given that the prior density, h (r;a',b'), is known and that the conditional density, g i. the posterior density, f Theorem: o I h.

$Lower Bayesian Confidence Limit, R, is defined by, L ( c* {^w R = i-a where R is the reliability of the component. Therefore, R. L(tt the Qfth percentile point of the posterior distribution of R.$

17 (rjs), (s r), R ) ) II. DISCUSSION A. BAYESIAN RELIABILITY Given that the prior density, h (r;a',b'), is known and that the conditional density, g i. the posterior density, f Theorem: o I h.s been determined by testing K can be computed by use of Bayes f Rls( r,s ) = I g s R (slq h (r ;a -,b-) R is (s r) h R (r;a',b')dr Once the posterior density has been determined the loo(l-a)! Lower Bayesian Confidence Limit, R, is defined by, L ( c* {^w R = i-a where R is the reliability of the component. Therefore, R. L(tt the Qfth percentile point of the posterior distribution of R.. i- c B. APPROXIMATING A BINOMIAL DISTRIBUTION WITH A BETA DISTRIBUTION The following identity relates the binomial distribution with the beta distribution. If U is distributed beta with parameters s and (n - s + 1), where n and s are positive integers and s ^ n, p(u^p) = s =s (jjp j (i-p) n ~ j, for ^ p ^ 1. Therefore, P(U ^ p) = P(X^ s) where X is distributed binomial with parameters n and p.

$r) h s f r s R s( = R (r;a«,b<), > ~ri g (s ) h (r;a',b')d* s R R a'-l,, *J* (1-r) r(a )r(bl) r,,b'-l 1/n^ ( x r\,_.n-s Ha'+b') a'-l/. N b'-1 a'+s-l..,.b'+n-s-l ('. a'+s-l..,.b'+n-s-l, (1-r) dr a'+s-l,,.$

19 R (r s) (s r), C. THE POSTERIOR DISTRIBUTION Given that the prior density, h (r;a',b') is beta with parameters a' and b' then h r (r ; a',b.) = ^T^y r a '- 1 (l-r) b '- 1 > for * r * 1 The conditional density, g i n and r where is binomial with parameters n = the number of items to be mission tested r = the probability of success s = the number of successes observed. Therefore, the conditional density is S s, R (s r). The posterior density, f i is given by g (s!r) h s f r s R s( = R (r;a«,b<), > ~ri g (s ) h (r;a',b')d* s R R a'-l,, *J* (1-r) r(a )r(bl) r,,b'-l 1/n^ ( x r\,_.n-s Ha'+b') a'-l/. N b'-1 a'+s-l..,.b'+n-s-l ('. a'+s-l..,.b'+n-s-l, (1-r) dr a'+s-l,,.b'+n-s-l 1 (*-*) r(a'+s)r(b'+n-s) r(a'+b'+n)

21 _ F a ( ' +b ' +n ) a'+s-l.,.b'+n-s-l s = l r (1-*) r(a +s)r(b»+n-s) T Therefore, the posterior density is beta with parameters (a' + s) and (b f + n - s). Let a = a' + s and b = b' +n - s. Therefore, the posterior distribution of R is beta with parameters a and b.

$, where u = 2 Arcsin up 2 11 \i and and = 1 j 2 Arcssin i y+ 2n p (x) y ex P Y (y) *" * 2 Arcsin \ + - 2 Arcsin «p 1 Where is a correction for continuity analogous to that used with the normal$

23 X. III. THE APPROXIMATION FORMULA A. DERIVATION If X is binomial with parameters n and p, then Y = 2 Arcsin J 0" 0" in radians is approximately normally distributed with parameters?,., where u = 2 Arcsin up 2 11 \i and and = 1 j 2 Arcssin i y+ 2n p (x) y ex P Y (y) *" * 2 Arcsin \ Arcsin «p 1 Where is a correction for continuity analogous to that used with the normal approximation. If U is distributed beta with parameters s and (n - s + 1), then n /n\. P [ u P ] = s [j) p D (i-p) n ~ 3 j^s = plx * s ] and P [U * p ] where X is binomial with parameters n and p. p[u ^ p] = P = P[2 Arcs in J - ^ 2 Arcsin J - in 3 II n = P[z ^(2 Arcsin j"^- 2 Arcsin J~p) \TrT ] Brownlee, K.A., Statistical Theory and Methodology in Science and Engineering, p , Wiley, I960 10

$1 - Where p = E X ] and Z is distributed normal (0,1). Now use ~ as n \ > j 2n a continuity correction factor.$ $2s + 1 2a + 1 n 2n 2n 2(a + b - 1) a + b -1 therefore, : p[u * p] = P Z ^{2 Arcsin J"p~ - ( and a 2 = ] 1 n a+b-1 2 Arcsin \ a+b-1 a+b-1 Therefore, 2 Arcsin U is approximately normally distributed \$

25 1 - Where p = E X ] and Z is distributed normal (0,1). Now use ~ as n \ > j 2n a continuity correction factor. P[U ^ p] = P[Z^(2 Arcsin - + j- - 2 Arcs in fp) \jr n ] = P[z ^(2 Arcsin \[p~ - 2 Ar csiu ' s. 1 + ~ ) frt ] If U is beta with parameters a and b, where a = s and b = n - s + 1 b = n - s + 1 b = n - a + 1 n = a + b - 1 s. 2s + 1 2a + 1 n 2n 2n 2(a + b - 1) a + b -1 therefore, : p[u * p] = P Z ^{2 Arcsin J"p~ - ( and a 2 = ] 1 n a+b-1 2 Arcsin \ a+b-1 a+b-1 Therefore, 2 Arcsin U is approximately normally distributed \ a+b-1 ' a+b-1 J ' In Bayesian analysis we want p such that plu > p] = l-a, ^ a ^ 1. Then p is the lower 100(l-a)% Bayesian Confidence Interval on U, Therefore Plu > p] = P[z >(2 Arcsin -fp - 2 Arcsin j a ^_ ) ja+b-1 ] (2 Arcsin fp" - 2 Arcsin 'J ^j^ ) ^ a+b-1 = Z 1-Q, where Z n, is a number that p[z > Z- 1,] = l-a and can be looked l-a l-a- in any Standard Normal Table. up 11

$2 Arcs in "p - 2 Arcs in \H +jj Jja+b-1 = Z 2 Arcs in J~p" = '1-a \j~a+b-l + 2 Arcsin \j a+b-1 fp = Sin + Arcsin ( a+b-1 P = Sin 2\ a+b-1 + Arcsin \ a+b-1 For simplicity, the fraction 3g was dropped$ $j a+b-1 2 \ a+b-1 I 2 2(a+b-l) However, if b < 1.0 then > 1.0 and the Arcsin is not defined a+b-1 when the argument is greater than one.$

27 2 Arcs in "p - 2 Arcs in \H +jj Jja+b-1 = Z 2 Arcs in J~p" = '1-a \j~a+b-l + 2 Arcsin \j a+b-1 fp = Sin + Arcsin ( a+b-1 P = Sin 2\ a+b-1 + Arcsin \ a+b-1 For simplicity, the fraction 3g was dropped from the numerator of the second term. This had the effect of reducing the approximated value of p - To compensate for this reduction, a correction factor was added to the formula as shown below: Sin '1-a + Arcsin.j a+b-1 2 \ a+b-1 I 2 2(a+b-l) However, if b < 1.0 then > 1.0 and the Arcsin is not defined a+b-1 when the argument is greater than one. To prevent this problem, so that the approximation formula can be used for all values of b, the second term was modified by adding 1.0 to the denominator. After comparing the approximated values with the computed ones, the correction factor was also modified by multiplying the denomi nator by 3 instead of 2. This resulted in reducing the approximation to just below the computed value of the Oith percentile points for low values of b, instead of just above it, as it was found to be originally. This change also caused the approximated values to be more uniformly accurate throughout the range of b. 12

$Therefore, the final appr oximation formula for the ath percentile points of the beta distribution is: / z \1 2 p«4 [ sin (nsi + Arcsin i^j\ +^ B.$ As shown in Tables I, II and III, the approximation formula results in extremely accurate values for the percentile points of the beta distribution, especially for values of b less than or equal to

As shown in Tables I, II and III, the approximation formula results in extremely accurate values for the percentile points of the beta distribution, especially for values of b less than or equal to

29 Therefore, the final appr oximation formula for the ath percentile points of the beta distribution is: / z \1 2 p«4 [ sin (nsi + Arcsin i^j\ +^ B. ACCURACY RESULTS The Qftn percentile points generated by the approximation formula compare very favorably with the computed values which will 1 hereafter be referred to as the "exact ' values. As shown in Tables I, II and III, the approximation formula results in extremely accurate values for the percentile points of the beta distribution, especially for values of b less than or equal to five. Examination of the results indicates that the approximated value is less than the exact value when b values are less than 2.5 and greater when b values are greater than 2.5. Therefore, for values of b near 2.5, the approximation formula results in values which are very nearly the same as the exact values. The accuracy cf the approximation is also a function of the parameter a. As the value of a increases the approximation becomes even more accurate. After using the approximation formula to compute various percentile points over a wide range of a and b values, it was noted that the worst estimated value was less than three thousandths away from the exact value for values of a greater than 40. Therefore, the formula will produce values for the Oith percentile points of two parameter beta distributions with an accuracy which should be completely sufficient for most applications 13

$o rh m 3 <f VO r-s 00 00 oo oo OO ON ON w X On cn T-i CO <1- VO ro m oo rh o On ro <t -cr CM ON in o r-. in c co UN m r- rh -Cl- CM VO VO m Du r^ vo \T\ o <t- VO 00 ON o rh % <r VO!$

31 '.-- o ' TABLE I. Comparison of Approximate and Exact 5th Percentile Points for the Beta Distribution ON <r CM rh -T o r^ rh in ON H r- m r-~ ro ON r-. m CO CO r-. -t ro CM CM rh rh rh rh o O o o o o (J O o be rh o o o O O o o O O H Q o ON ON CM o vo VO <f vo VO O rh VO ON m VO m CO CM rh o U o OO o CO oo CM o <f in <r vo IfN m o ro vo OO ON o rh m 3 <f VO r-s oo oo OO ON ON w X On cn T-i CO <1- VO ro m oo rh o On ro <t -cr CM ON in o r-. in c co UN m r- rh -Cl- CM VO VO m Du r^ vo \T\ o <t- VO 00 ON o rh % <r VO!> oo 00 oo oo 00 ON ON VO CM oo CO ON rh t^ m!- r^ rh r^. CM vo o CM -t ro o <t CM O LfN ih O ON rh O vo o CO H rh o CM rh O o ON o be Q O o o o o o o o o o m IT m cm ON ON r^ in m o CO <t- CM <jr^ o m m CM vo CM <f CO O vo ON vo ro r>. ON rh CM CO <r <f a IT) t>. OO 00 O0 ON ON ON ON ON [X] rh ON vo ON r-. r-- 00 m X NO ih ro r». CO oo CM m ON ON o oo o VO -cr VO m rh ON 00 <r tx r^ rh 00 m oo -T VO <r rh i> n, o I\ co r^ ON rh CM CO <f <ivo l> OO oo 00 ON ON ON ON ON Si ON rh m CM <fr CM in rh rh CM o <r <f ON VO CO rh o ON t-- m co CM H rh rh rh rh It, O o o o o O O O o o o be Q ' 1 ' rh in CO ro o VO <t o CO CO H ro ro -.T oo CM CO 00 CM in ) t-l tv h» CM CM 00 CM VO 00 o CM O ro in VO vo r^ 1> r» OO o OO ON ON ON ON ON ON ON ON ON X CM -f CO rh vo <l" ON ON CM rh n CM ON o ON m o VO rh ON rh a VO CO in o o r^- rh m t^ o r\ t-! o ro in vo VO ^ l> «> 00 OO ON ON ON ON ON ON ON ON ON -/ o o o o o o o o o o / ih CM co <i- in VO r-«00 ON o / «H 14

33 TABLE II. Comparison of Approximate and Exact 10th Percentile Points for the Beta Distribution m '-- CM on in r- CO ~:f H rh <f o m CO <t CO H C7n oo [> -st VO CO CM CM r-t H O O o be H O O O be O o o o o o o (0 o o H Q B 3 W vo CM H 00 o r^ hs ON VO >-{ ON H in <l" r^ ON co o I"- VO o ON r^ CM in r^ ON o H CM o r>. <t co VO VO co <r h- VO <r m m VO r» 00 co CO oo ON ON ON X H ON co VO m <t (O r*-. CM O <t rh O CO H en o o in CO tt CM CM CO ON o VO 00 r-. m 0, CM O CO CM in CO ON o r-i CM m J> r-- CO CO CO 00 ON ON ON % CM CO U"\ in r-\ CM 0O o CO oo vo UN CO CO m -f <l- co CM o co r-i H O O u. r-i O O O a O be o o o o o o o o o o Q f + --f VO CO ro co o o oo ON m CM <r CO o oo oo <r CO r-i in ft o> O o CM co VO VO <r O m. -cr o VO ON rh CM CO <r in in cm S VO co oo CO ON ON ON ON ON ON tu X vo <f co CO rh o O0 o rh IN o ih CO ro ON co <f CM co i>. <Jtt o CO CM ro CO h» r-- -i- o in 0- VO o VO ON rh CM CO <t m m VO co CO co ON ON ON ON ON ON S I s * r>. m ro CM CO VO VO ON vo VO H r>. ON <f o h- in ro CM r- m CO CM CM CM r-< r-i rh rh be O O O O O O O be o o o O o o o o o o Q i i 1 i ' CM r-i IT -.1- o r-< ON ON <1- CO m CO m m CM VO h- rh o in lo R o ro in VO ro rv o CO m vo h» ro m VO r>. r*- oo CO oo co o co ON ON ON ON ON ON ON ON ON & / (0 *s^ X in <1- o rh CO CO CO CO in f> o 00 VO co VO r- UN O VO VO CM cc CM co rh CO O in ON r-i co in 0. VO CM m VO r- r-> r> CO 00 ON ON ON ON ON <J\ ON ON ON % o o o O o o o o o rh CM CO <r m VO r- co ON o rh 15

$Comparison of Approximate and Exact 20th Percentile Points for the Beta Distribution o ON VO ON <t <r in CM CO VO 00 CM VO U"\ o r» in <f CO CM <r in CM rh rh fcc H o o o o o o o o o 1-1 Q O o o o o$ $o o o o o + + + + + + + + + + CO r^ H VO r^ rh <t CO CO CM CM r*. CM ON VO CNJ 00 CM in cm o B vo m O rh CO is o i ON VO vo ro rh in r-- ON rh CM CM CO m lo I-- OO oo oo 00 ON o\ ON ON W X OO vo.$

35 1 : TABLE III. Comparison of Approximate and Exact 20th Percentile Points for the Beta Distribution o ON VO ON <t <r in CM CO VO 00 CM VO U"\ o r» in <f CO CM <r in CM rh rh fcc H o o o o o o o o o 1-1 Q O o o o o o o o o o CO r^ H VO r^ rh <t CO CO CM CM r*. CM ON VO CNJ 00 CM in cm o B vo m O rh CO is o i ON VO vo ro rh in r-- ON rh CM CM CO m lo I-- OO oo oo 00 ON o\ ON ON W X OO vo.s in H in ON o VO 00 o O o oo in r-- ON ro r» oo <ftx H rh CM CO ON Is - rh r-l ON VO 0. OO <f rh in hs ON rh CM CM CO m IS OO oo 00 OO ON On ON ON 3= h- OO o VO CM rh in On O rh ON -cf ON ro rh c rh rh ON CM u. O (x, M o O o o o o o o o o 1 Q CM VO 00 CM On VO oo is ro C»s m 1^ rh <f VO r-- Is a) VO rh B rf IT rh 00 oo o rh ON UN m o o- CNJ x H fv ro OO oo oo rh ON CO ON <t ON <r On ON VO ON vo ON X On <f oo OO rh s- CO a) CO VO o <t- CM o I s -. CO h- s- is in ro a rh rh ON <r O rh ON in o <r 0, CM <r oo H co <1" <f m VO vo 8! IS ON ON ON ON ON ON On ro ro oo oo rh o CM ON <r cm O <f CM m rh oo m CO CM rh vo -T CO CM CM rh rh rh rh H fx, {i, O o O o o O o o O o o o Q ' ' 1 ' 1 1 ' 1 1 1». OO oo m CM m m VO On O in h ON ON CM ON CO VO oo ON O H o w rh ON LfN ON r>. ON s I - ON oo ON CO ON oo ON 00 ON ON ON ON ON CM CM s~ m vo ro CM is oo oo X <t m o s. rh m CO IS m oo n CM oo in ON in in I s - CO VO vo rx co <r C7N VO H <t VO CO On O n. rh m VO is co CO 00 CO OO ON ON ON ON On On ON ON ON On ON k */ o o o o O o o a o o / H CM en <i- m VO is co ON O / «rh -16

$'l-a 2 \ a+b-l + Arcsin a+b 2(a+b-l the resulting percentile points will all be greater than the exact values. The accuracy of the approximated values is extremely good for b ^ 1.$

37 { ) This is especially true for values of the 20th percentile points, Table III, for which the approximated values are exceptionally accurate. C. ALTERNATE APPROXIMATION FORMULAS If the approximation formula is used with the continuity correction factor that was originally derived, P = Sin / Z. 'l-a 2 \ a+b-l + Arcsin a+b 2(a+b-l the resulting percentile points will all be greater than the exact values. The accuracy of the approximated values is extremely good for b ^ 1.0, but as the value of b increases, the approximated values exceed the exact values by an ever increasing amount. As a result, the approximated values become unacceptable when compared with the values generated by the formula using. as o a+b-l the continuity correction factor. See Table IV for a comparison of the values generated by the two formulas. When the continuity correction factor is decreased by using 4(a+b-l) for the denominator, the approximated values will be less than the exact values. This trait could be useful if conservative estimates were desired. The approximation formula P = Sin 'l-a 2 y a+b-i + Arcsin a+b. 4 (a+b-l) is most accurate for values of b ^ 4.0 due to the fact that, as the value of b increases, the approximated value of p increases 17

. with respect to the^exact value. However, the approximated percentile points for lower values or" b are not nearly as accurate as those given by the chosen formula.

$0 from the denominators of the first term and the correction factor. The resulting formula, / z \1 2 Sin + Arcsin \ * V2faTb a+b /J -r- / + _, jv 3U b).$

39 . with respect to the^exact value. However, the approximated percentile points for lower values or" b are not nearly as accurate as those given by the chosen formula. See Table IV for a comparison of the values generated by the two formulas. The appr oximation formula can be simplified by not subtractirg 1.0 from the denominators of the first term and the correction factor. The resulting formula, / z \1 2 Sin + Arcsin \ * V2faTb a+b /J -r- / + _, jv 3U b). gives very accurate approximations of the percentile points and is quite acceptable for most applications. This formula is most accurate for b ^ 2.5 and should be used whenever b is restricted to these values. The chosen formula is only superior in that it results in extremely accurate estimates for all values of b, not just very low values. However, the formula presented above would actually be preferred whenever b is restricted to very low values. See Table IV for a comparison of the values generated by the two formulas 18

.5 Exact TABLE IV. Comparison of Approximate and Exact 20th Percentile Points Generated by Selected Approx: mat ion Formulas a b for the Beta Distribution F-l F-2 F-3 F-4 50.5.98362.98151.98487.97982.

90933.91250.90774.90953 50 4.o.89840.89915.90229.89758.89936 50 4.5.88838.88929.89240.88773.88951 50 5.o.87867 87971.88280.87817.87995 ioo.99180.99068. 99236.98984.99069 100 1.0.98403.98337.98504.

41 .5 Exact TABLE IV. Comparison of Approximate and Exact 20th Percentile Points Generated by Selected Approx: mat ion Formulas a b for the Beta Distribution F-l F-2 F-3 F ' o o ioo F-l p = / v 2 z l-a J a+b-1 + Arcsi f a+b 3(a+b-l F-2 p = F-3 P = Sin ( z l-a f Sin/ {? \ a+b-1 I i2 ^ z l-tt a+b-1 atbj \' + Arcsin ^ 2(a+b-l + 4(a+b-l) F-4 p = Sin j / Z l-<* 2fa+b U S+b 3(a+b) 19

$n = 50.0 and s = 49.0 a = a' + s = 51.0 + 49.0 = 100.0 b = b T +n-s = 1.0 + 50.0-49-0 = 2.0 a. =.20 and Z 1 _Q, = Z ^Q = -.842 "l-a Sin j + Arcsm 2 ^a+b-1 \ a+l 3(a+b-l) 20 -.842. [~Too\ r==r + Arcs in J in0+?$ 2^ 100+2-1 ' 100+2y 3(100+2-1) = [Sin (-.04189 + 1.43031)] +.00330 = (.98341) 2 +.OO33O =.96709 +.00330 =.97039 The table value is.97064 and the difference is only -.00025. B.

2^ 100+2-1 ' 100+2y 3(100+2-1) = [Sin (-.04189 + 1.43031)] +.00330 = (.98341) 2 +.OO33O =.96709 +.00330 =.97039 The table value is.97064 and the difference is only -.00025. B.

43 IV. EXAMPLES A. EXAMPLE 1 Find the 20th percentile point for a = and b = 2.0. Assume a' = 51.0 and b* = 1.0. A random sample of 50 items was then mission tested and one failure was observed. n = 50.0 and s = 49.0 a = a' + s = = b = b T +n-s = = 2.0 a. =.20 and Z 1 _Q, = Z ^Q = "l-a Sin j + Arcsm 2 ^a+b-1 \ a+l 3(a+b-l) [~Too\ r==r + Arcs in J in0+? 2^ ' 100+2y 3( ) = [Sin ( )] = (.98341) 2 +.OO33O = = The table value is and the difference is only B. EXAMPLE 2 Find the 25th percentile point for a = 50.0 and b = 2.5. Assume a' = 26.0 and b f = 1.5. A random sample of 25 items was then mission tested and one failure was observed. n = 25.0 and s = 24-0 a = a' + s = =

5 T b=b +n - s =1.5+25.0-24.0 = 2. d =.25 and Z 1 _Q, = Z <?5 = -.675 Sm / 1-0?

5-1 Sm o / 3(50+2.5-1) 2 = [Sin (-.04703 + 1.35088)]" +.00647 = (.96456) 2 +.

45 5 T b=b +n - s = = 2. d =.25 and Z 1 _Q, = Z <?5 = Sm / 1-0?..a + Arcsn 2 i a+b-1 «a+b 3(a+b-l' 25 / ' + Arcsm ^2 j Sm o / 3( ) 2 = [Sin ( )]" = (.96456) = = The table value is and the difference is only , 23

BIBLIOGRAPHY Cl] Brown] ee, K.A., Statistical Theory and Methodology in Science and Engineering, p. 115-116, Wiley, I960. [2] Lindqren, B.W., Statistical Theor y, 2d ed.

47 BIBLIOGRAPHY Cl] Brown] ee, K.A., Statistical Theory and Methodology in Science and Engineering, p , Wiley, I960. [2] Lindqren, B.W., Statistical Theor y, 2d ed., p ; 3/0-373, Macmillan, [3] Zehna, P.W., Probability Distributions and Statistics, p , , Allyn and Bacon,

INITIAL DISTRIBUTION LIST No. Copies 1. Defense Documentation Center 2 Cameron Station Alexandria, Virginia 22314 2.

Library, Code 0212 2 Naval Postgraduate School Mo

49 INITIAL DISTRIBUTION LIST No. Copies 1. Defense Documentation Center 2 Cameron Station Alexandria, Virginia Department of Operations Analysis (Code 55) 1 Naval Postgraduate School Monterey, California Library, Code Naval Postgraduate School Monterey, California Prof. W.M. Woods, Code 55 Wo 3 Department of Operations Analysis Naval Postgraduate School Monterey, California Capt. Thornton Boyd, USMC Chalcedony Street San Diego, California

(Type of report end. inc lustve dates) Master's Thesis; March 1971 S»utmORi5i (firsi rumf, middle initial, laat name) Thornton Boyd REPORT DATE March 1971 CONTRACT OH GRANT NO 7a. TOTAL NO.

51 Unclassified DOCUMENT CONTROL DATA R & D - fastifif afion of title, body i,l abstract and indexing annothti :.:. Naval Postgraduate School Monterey, California I E POR T SECURITY C Unclassified thod of Approximating Percentile Points of Highly Skewed, Two Parameter Beta Distribution descriptive: NOTES (Type of report end. inc lustve dates) Master's Thesis; March 1971 S»utmORi5i (firsi rumf, middle initial, laat name) Thornton Boyd REPORT DATE March 1971 CONTRACT OH GRANT NO 7a. TOTAL NO. OF PACES 25 ORKIM*TOR'S REPORT lb. NO. OF KEI 3 b. PROJEC T NO DISTRIBUTION STATEMFNT Approved for public release; distribution unlimited. 5UPPL EMEN T AR> 12. SPONSOI ILITARV ACTI Naval Postgraduate School Monterey, California An approximation formula is derived which provides a simplified and efficient method of calculating the Q!th percentile point of a two parameter beta functioi when the two parameters are known. A specific application is presented by utilizing the formula to compute the lower 100(l-a)% Bayesian confidence limit for the reliability of a component when the prior distribution of that reliability is known to be beta with parameters a' and b'. The posterior distribution is then determined by mission testing n items and recording the number of successes, s. This distribution is known to be beta with parameters a and b, where a = a' + s and b = b' + n - s. Therefore, the formula can be utilized to determine the ath percentile point of this posterior distribution which by definition is the lower 100(1-0.')% confidence limit for the reliability of the component. 1PAGE DD ".'..V Unclassified S/N Security Classification

53 I Unclassif i od KEY WORDS Bayesian Reliability Beta Distribution Percentile Points Posterior Distribution DD. F.T 1473 'back) 25 Unclassif ied S/N B? Security Classification

2515 Thesis * 7 C ' f 125922 Y A method of approx- '.

stributions. 225 15 Thesis 125922 B792 Boyd c.

57 2515 Thesis * 7 C ' f Y A method of approx- '.mating percent, le points of highly skewed, two parameter beta d.stributions Thesis B792 Boyd c.l A method of approximating percenti le points of highly skewed, two parameter beta distributions.

58 thesb792 A method of approximating percentile IIIISI DUDLEY KNOX LIBRARY

UNCLASSIFIED UNCLASSIFIED DEFENSE DOCUMENTATION CENTER SCIENTIFIC AND TECHNICAL INFORMATION FOR CAMERON STATION ALEXANDRIA VIRGINIA

UNCLASSIFIED UNCLASSIFIED DEFENSE DOCUMENTATION CENTER SCIENTIFIC AND TECHNICAL INFORMATION FOR CAMERON STATION ALEXANDRIA VIRGINIA UNCLASSIFIED AD DEFENSE DOCUMENTATION CENTER FOR SCIENTIFIC AND TECHNICAL INFORMATION CAMERON STATION ALEXANDRIA VIRGINIA DOWHGRADED AT 3 TOAR INTERVALS: DECLASSIFIED ATTER 12 YEARS DOD DIR 520010 UNCLASSIFIED