The inverted beta loss function: properties and ... - Springer Link

IIE Transactions (2002) 34, 1101–1109

The inverted beta loss function: properties and applications BARTHOLOMEW P.K. LEUNG1 and FRED A. SPIRING2 1

Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong Department of Statistics, The University of Manitoba, Winnipeg, Manitoba, Canada R3T 2N2 E-mail: [email protected]

2

Received July 2000 and accepted May 2002

In this paper, a general class of loss functions based on the inversion of the standard beta probability density function (pdf) is examined. The extension of this loss function from a standard beta pdf ranging from (0, 1) to the general beta pdf ranging from (p, q) is examined through the scale invariance property under a linear transformation. An industrial application in quality assurance is used to demonstrate this general class of loss functions. Mathematical derivations are attached in the Appendices.

1. Introduction The loss function approach for assessing quality was first motivated by Taguchi (1986) who used a modified quadratic loss function to assess and illustrate losses associated with departures from a process target. Much of the recent practical loss function work has been motivated by either enhancements to existing loss function material or practical problems that could not be adequately dealt with using existing published material. Spiring (1993) proposed an Inverted Normal Loss Function (INLF), which differed from traditional quadratic loss by providing a bounded, and hence more reasonable assessment of economic loss. Suggesting that the INLF severely penalizes off-targetness, Sun et al. (1996) developed a modified INLF that provided a more moderate loss representation. They also provided a method for fitting the modified INLF to reflect the user’s actual loss function. The result was a nonlinear least squares method for estimating the shape parameter of their modified INLF. This was an important step in facilitating the use of the modified INLF, as quadratic loss requires little in the way of fit. As pointed out by an anonymous referee ‘. . . the quadratic approach requires only the determination of a constant’ as well as ‘estimates of the process mean and variance’ to describe loss. In addition ‘No distributional assumptions are necessary’. The ability to fit the modified INLF to reflect the user’s actual loss moved the INLF from a research curiosity to a usable method in the area of loss. Spiring and Yeung (1998) developed a class of loss functions based on inverted probability density functions

0740-817X

Ó 2002 ‘‘IIE’’

(pdfs) including the gamma, Tukey’s Symmetric Lambda and Laplace distributions. This general class of loss functions has attractive properties and can accurately reflect symmetric and asymmetric losses incurred by the process as suggested in their set of examples. However the various loss functions discussed have parameters nested in their associated pdfs. In this paper we first develop a family of symmetric and asymmetric loss functions based on an inverted beta pdf. The approach used in the development is consistent with Spiring and Yeung (1998) allowing us to then describe the risk function associated with the proposed family of Inverted Beta Loss Functions (IBLFs). A method for fitting/selecting an appropriate IBLF is outlined followed by a section devoted to the physical and statistical properties of the inverted beta loss function. The results are then illustrated through an example from the printing industry. The shape of the IBLF can be modified to suit the practitioner’s needs, while providing all the properties of the above mentioned loss functions. By restricting our family of loss functions to those derived from the beta pdf we manage to provide a wide variety of potential loss functions while maintaining a single set of parameters for the entire family. The family of IBLFs, while adding flexibility to the shape that loss functions can attain, is more complex than the quadratic loss in attaining a suitable fit to the actual loss incurred. However similar to Sun et al. (1996), a method for fitting the IBLF to actual losses incurred is included. The loss and risk assessment of an IBLF can be evaluated on a process characteristic having finite or infinite support by means of a

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Leung and Spiring

transformation. All figures are plotted using Mathematica, version 2.2.3 (Wolfram, 1991).

0
0, b > 0 having the functional form 1 xa1 ð1  xÞb1 ; Bða; bÞ

f ðxÞ ¼

0 < x < 1;

the modal point occurs at MO ¼

a1 ; aþb2

which represents the value of x at which the beta pdf attains its maximum. To insure the existence of a unique maximum we must have a > 1, b > 1 because @ 2 ln f ðxÞ a1 b1 ¼ 2 þ ; evaluated at 2 @x x ð1  xÞ2 a1 ; 0 < MO < 1; MO ¼ aþb2 ¼

ða þ b  2Þ3 1 and b > 1: Let T denote the ideal target of the process and assume T ¼

a1 ; aþb2

to be fixed. Using the unique maximum conditions associated with the beta distribution, a linear relationship can be established between a and b through T. The relationship can be written as T 1  2T T ðb  1Þ b þ and a  1 ¼ : 1T 1T 1T Letting pðx; T Þ denote a function of the form (a beta probability density function) a¼

1 pðx; T Þ ¼ xa1 ð1  xÞb1 ; Bða; bÞ

0 < x < 1;

The loss function is used to describe the loss incurred when a process departs from the target. In decision theory the risk function provides the average loss associated with the process given the loss function and some assumed distribution for the process measurements. It provides an assessment of the average loss to customers or society when the target is missed. The IBLF is chosen to fit the practitioners’ need and its associated risk function can be evaluated easily. In particular if the process characteristic X has a standard beta distribution with parameters aR > 0 and bR > 0, then the expected loss or risk associated with the IBLF is E½LðX ; T Þ ¼

Z1

 h iða1Þ  K 1  C xð1  xÞð1T Þ=T

0

1 xaR 1 ð1  xÞbR 1 dx; BðaR ; bR Þ 0 1 Z1 C ðð1T Þ=T Þða1Þþb 1 ðaþa 1Þ1 R R ¼ K @1  x ð1  xÞ dxA; BðaR ; bR Þ 0   Bða þ aR  1; ðð1  T Þ=T Þða  1Þ þ bR Þ ¼K 1C : BðaR ; bR Þ

with m ¼ sup pðx; T Þ ¼ x

1 T a1 ð1  T Þb1 ; Bða; bÞ

where m denotes the supremum of pðx; T Þ. Then analogous to Spiring and Yeung (1998), the loss inversion ratio becomes

ð3Þ

1103

Inverted beta loss function

partial information regarding the actual loss associated with a deviation from T are known. The most frequent case that occurs includes the situation where the maximum loss (and its first occurrence) and the loss at T (i.e., zero) are known. This is referred to as ‘the primary loss information’. If additional information is available, it can be used to provide a superior representation of the loss function, keeping in mind that the goal is to accurately depict the losses associated with deviations from the target (T). In those cases where only the ‘primary loss information’ is specified, i.e., T and K, where 0 < T < 1, the general form of the IBLF is  pðx; T Þ Lðx; T Þ ¼ K 1  ; 0 < x < 1; m where K is the maximum loss, pðx; T Þ a beta pdf with parameters a and b, T ¼

a1 ; aþb2

is the target of the process, m is the supremum of p(x, T). The associated risk function is Equation (3) when assuming X Be (aR , bR ). The shape of this loss function can be controlled through the selection of a and/or b as long as both a > 1 and b > 1. Since a, b and T are related as follows: T ðb  1Þ ð1  T Þða  1Þ or b  1 ¼ ; ð4Þ 1T T and assuming T to be fixed, there are many combinations of a and b which can be used to create various shapes for the IBLF (see Fig. 1(a–c)). When T ¼ 1=2, from Equation (4) it is easy to verify that a ¼ b. Assuming T ¼ 1=2 and a ¼ b > 1, the resulting IBLF is symmetric around T with the maximum loss reached at similar distances from T in both directions. Figure 2 illustrates three of the many symmetric forms the IBLF may take on. a1¼

Fig. 1. (a): (i) L1 ðx; T ¼ 0:15, a ¼ 1:5, b ¼ 3:83Þ, (ii) L2 ðx; T ¼ 0:45, a ¼ 1:5, b ¼ 1:61Þ, (iii) L3 ðx; T ¼ 0:80, a ¼ 1:5, b ¼ 1:125Þ; (b): (i) L1 ðx; T ¼ 0:15, a ¼ 5, b ¼ 23:67Þ, (ii) L2 ðx; T ¼ 0:45, a ¼ 5, b ¼ 5:89Þ, (iii) L3 ðx; T ¼ 0:8, a ¼ 5, b ¼ 2:00Þ; and (c): (i) L1 ðx; T ¼ 0:15, a ¼ 20, b ¼ 108:67Þ, (ii) L2 ðx; T ¼ 0:45, a ¼ 20, b ¼ 24:22), (iii) L3 ðx; T ¼ 0:8, a ¼ 20, b ¼ 5:75Þ.

The beta distribution provides a closed form for the risk function associated with the IBLF. An application illustrating this property will be presented through an example taken from the printing industry. However there are other distributions, which can be used with the IBLF and these will be discussed at later points of this paper.

4. Choosing an appropriate IBLF In practice, loss functions are chosen to reflect the loss associated with processes that vary. In many cases only

Fig. 2. (i) L1 ðx; T ¼ 0:5; a ¼ 1:5; b ¼ 1:5Þ, (ii) L2 ðx; T ¼ 0:5; a ¼ 1:5; b ¼ 1), and (iii) L3 ðx; T ¼ 0:5; a ¼ 1:5; b ¼ 1:5Þ.

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Leung and Spiring

When T 6¼ 1=2, the IBLF will be asymmetric and have many potential shapes as shown in Fig. 1(a–c). From Fig. 1(a–c) and Fig. 2 we see small values of a ‘open up the arms’ of the loss function around T, while larger a values ‘tighten the arms around T ’. Small a values result in smaller economic losses for slightly off target processes, while larger values of a assign a more severe penalty (loss) for similar departures from the target. In those cases where T, K and an auxiliary piece of information about the loss are known (e.g., [x1 , L1 ], where L1 represents the loss at x1 ), the value of a is such that  pðx1 ; T Þ ; ð5Þ L1 ðx1 ; T Þ ¼ K 1  m while continuing to satisfy the conditions outlined in Equation (4) and a can be solved using Equation (5) using Ki ¼ K for i ¼ 1, 2. The associated risk function will be the same as that described in Equation (3). It can apply whenever x1 < T or x1 > T . The situation is shown in Fig. 3(a and b). When T, K and two auxiliary pieces of information (e.g., ½x1 ; L1  and ½x2 ; L2 ) are known and x1 < T and x2 > T , we need to solve for a1 and a2 such that  p1 ðx1 ; T Þ and L1 ðx1 ; T Þ ¼ K 1  m1  p2 ðx2 ; T Þ : L2 ðx2 ; T Þ ¼ K 1  m2 It is easy to show the solutions are of the form #!   , "  Li ðxi ; T Þ xi 1  xi ð1T =T Þ ai ¼ ln 1  ln þ 1; Ki T 1T i ¼ 1; 2;

Fig. 3. (a) T ¼ 0:75, K ¼ 10, L½0:6; 0:75 ¼ 4, a ¼ 8:6844, b ¼ 3:5615; and (b) T ¼ 0:75, K ¼ 15, L½0:85; 0:75 ¼ 6, a ¼ 12:3235, b ¼ 4:7745.

ð6Þ

where L1 and L2 represent the losses associated with the values of the process characteristic x1 and x2 respectively. Combining the resulting curves provides practitioners with a versatile loss function of the form  8
< K 1  p1 ðx;T Þ m1
 if 0 < x < T ; Lðx; T Þ ¼ ð7Þ p ðx;T Þ 2 if T < x < 1; : K 1 m2

allowing both sides of the target to have a maximum loss of K and shape based on p1 ðx; T Þ (which has parameters a1 and T ) and p2 ðx; T Þ (which has parameters a2 and T ). Figure 4 illustrates the combined loss function based on two different beta pdfs. The associated risk function, assuming X Be (aR , bR ), can be shown to be E½LðX ; T Þ ¼ Kf1  C1 BT ðl1 þ aR ; n1 þ bR Þ=BðaR ; bR Þ  C2 ½Bðl2 þ aR ; n2 þ bR Þ  BT ðl2 þ aR ; n2 þ bR Þ=BðaR ; bR Þg; where

Fig. 4. T ¼ 0:4, K ¼ 2:5, L1 ð0:3; 0:4Þ ¼ 1:25, and L2 ð0:75; 0:4Þ ¼ 1:2.

h i1ai Ci ¼ T ð1  T Þð1T Þ=T ; li ¼ ai  1; 1T ðai  1Þ; i ¼ 1; 2; T RT and BT ða; bÞ ¼ 0 xa1 ð1  xÞb1 dx, the incomplete beta function. ni ¼

1105

Inverted beta loss function

Fig. 5. T ¼ 0:65, K1 ¼ 15, L1 ð0:2; 0:65Þ ¼ 5, K2 ¼ 20, and L2 ð0:7; 0:65Þ ¼ 5:

For those situations where either the maximum is different on either side of the target the IBLF can be combined as follows:  8
< K1 1  p1 ðx;T Þ m1
 if 0 < x < T ; Lðx; T Þ ¼ ð9Þ p ðx;T Þ 2 if T < x < 1; : K2 1  m2 allowing either side of the target to have maximum losses of K1 and K2 respectively and a shape based on p1 ðx; T Þ and p2 ðx; T Þ. See Fig. 5 for an example. The corresponding risk function, assuming X Be (aR , bR ), can be shown as follows: E½LðX ; T Þ ¼K1 fIT ðaR ; bR Þ  C1 BT ðl1 þ aR ; n1 þ bR Þ =BðaR ; bR Þg þ K2 f½1  IT ðaR ; bR Þ  C2 ½Bðl2 þ aR ; n2 þ bR Þ  BT ðl2 þ aR ; n2 þ bR Þ=BðaR ; bR Þg; ð10Þ where IT ða; bÞ ¼

ZT

1 xa1 ð1  xÞb1 dx; Bða; bÞ

0

is the incomplete beta function. Equation (10) reduces to Equation (8) when K1 ¼ K2 ¼ K and this point will be discussed at a later point of this paper.

Fig. 6. (a) Lðy; T 0 ¼ 33, a ¼ 2, b ¼ 1:54); and (b) Lðx; T ¼ 0:65, a ¼ 2, b ¼ 1:54Þ:

mode is independent of scale. It follows that ‘one minus this ratio’ is also independent of scale, and hence the IBLF is said to be scale invariant under linear transformations. This shows the flexibility of an IBLF constructed using a standard beta pdf on (0, 1) to a generalized IBLF on (p ¼ a, q ¼ a þ b). To illustrate, Fig. 6(a and b) contains the IBLF associated with the standard beta pdf and K ¼ 10 (i.e., Lðx; T ¼ 0:65Þ) and the IBLF associated with the transformation y ¼ 20 þ 20x again with K ¼ 10 (i.e., IBLF Lðy; T ¼ 33Þ) (see Appendix A for proof). Figure 6(a and b) are IBLF’s with K ¼ 10. The risk function is scale invariant under a linear transformation. It follows directly from the property of expectation (see Appendix B for proof). The risk function associated with the IBLF has a closed form for all distributions with finite moments. The general form of the risk function for the IBLF is E½X l ð1  X Þn  and can be evaluated for all cases where the moments exist (see Appendix C for proof).

5. Some properties of the IBLF 6. An application of the IBLF The shape of the IBLF is scale invariant under a linear transformation. If the IBLF is based on a generalized beta distribution (i.e., f ðxÞ) with unique maximum conditions, then a transformation of the form y ¼ a þ bx results in an IBLF with a similar shape but a different scale and/or target. Assuming f ðxÞ to be a standard beta pdf with unique maximum conditions, then the transformation y ¼ a þ bx results in ymax ¼ a þ bxmax . As a result the ratio is also independent of the pdf so its unique

A lottery ticket manufacturer produces tickets that are distributed through vending machines. The tickets are to be folded and stacked in columns within the vending machine and dispensed one at a time through a dispensing slot. After inserting sufficient funds, a ticket is exposed and the purchaser is required to tear the ticket from the dispenser. The vending machine operators identified the critical characteristic in this process as the force

1106 required to remove the ticket from the dispenser. This force was directly related to the ‘pull strength’ associated with perforations made to the ticket during manufacturing. From past experience, the vending machine operators found that when the perforation pull strength was above 60 pounds per square inch (psi), tickets would not necessarily break along the perforation, leaving portions of the ticket inside the vending machine. It was also found that in those cases where pull strengths were less than 40 psi, the vending machine tended to supply more than one ticket at a time. This resulted in the vending machine jamming as the next ticket would not feed properly through the mechanism. In the case of a vending machine jam the company felt the cost to restore the machine to working condition was $0.10 per ticket. If the pull strength was beyond 60 psi, the vending machine was unable to break the perforation cleanly and the loss per ticket was also considered to be $0.10. The manufacturer agreed to compensate the vending machine company on a sliding scale that accurately reflected the costs associated with off-target perforations. Both parties agreed that $0.10 per ticket fairly depicted the costs associated with a complete failure of the perforation and that this occurred when the perforation strengths were outside the 40–60 psi interval. In addition they both agreed that the scale must include a $0.05 per ticket penalty to the manufacturer if the pull strength were 45 or 57.5 psi. Using this information, an IBLF was ultimately used to reflect the compensatory package for perforations that were off target. In addition the manufacturer was interested in determining their risk exposure under normal operating conditions. Using Equation (6), with T ¼ 55, K ¼ 0:1, x1 ¼ 45, L1 ðx1 ; T Þ ¼ 0:05, a1 ¼ 1:9464 and x2 ¼ 57:5, L2 ðx2 ; T Þ ¼ 0:05, a2 ¼ 10:0138, results in the IBLF illustrated in Fig. 7. The original and transformed Y  40 X ¼ 60  40

Fig. 7. L1 ðx1 ¼ 45, T ¼ 55, K ¼ 0:1Þ ¼ 0:05, a1 ¼ 1:9464 and L2 ðx2 ¼ 57:5, T ¼ 55, K ¼ 0:1Þ ¼ 0:05, a2 ¼ 10:0138:

Leung and Spiring i.e., p ¼ 40, q ¼ 60, pull strength data are included in Appendix D. The pull strength data appears to follow a beta [aR ¼ 2:0994, bR ¼ 2:3184] (verified by the chisquare goodness-of-fit test (p-value ¼ 0:2638)). The expected loss was calculated using Equation (8) and turned out to be $0.028. When presented in this fashion, it was possible to better assess the impact of the problem. The numerical results, in particular the expected loss of $0.028 per ticket, permitted decisions regarding pricing and quality improvement strategies to be made using tangible information.

7. Summary and conclusions The IBLF is easy to construct as the various choices of a, for a fixed target T, allows the IBLF to be tailored to the practitioners’ need. In general the relationship that exists among T, a and b suggests that for a fixed T, as a increases, b will increase. Alternatively, when keeping a fixed and increasing T, b will decrease. It can be shown that when T ¼ 1=2, we need not have a ¼ b, for two auxiliary pieces of loss information are used, permitting asymmetric shaped loss functions. Alternatively we can use Equations (7) or (9) to allow the loss function to have asymmetric shapes. Figure 8(a and b) illustrates two IBLFs with T ¼ 1=2 but a 6¼ b and various values of K. The general form of the IBLF admits a closed-form risk function for those distributions having finite moments. The expectation of Lðx; T Þ can be obtained easily even if the loss function is a combination of two different loss functions. In particular, if the beta distribution is employed as the conjugate distribution (closed under

Fig. 8. (a) L1 ðx; K1 ¼ 18, a1 ¼ 10Þ, L2 ðx; K2 ¼ 18, a2 ¼ 5Þ; and (b) L1 ðx, K1 ¼ 15, a1 ¼ 5Þ, L2 ðx, K2 ¼ 20, a2 ¼ 3Þ:

1107

Inverted beta loss function sampling), Equations (8) and (9) are the solutions. If the process measurements follow a distribution of the form of a normal, gamma, Weibull, . . . etc., the risk function still can be found. Under a linear transformation, the IBLF and its risk function are the same as those IBLF associated with the standard beta pdf. This follows directly from the result that a generalized beta distribution can be transformed to a standard beta distribution. If IBLF is used to assess the loss of a process having a normal distribution with mean l and variance r2 , then we may generalize the IBLF with p ¼ l  kr and q ¼ l þ kr, with k  3:5 so as to cover almost the range of the process measurement. In the case where the process characteristic follows a gamma distribution withpparameters a and b, then set p ¼ 0 and ffiffiffi q ¼ ab þ kb a, i.e., mean þ k standard deviations of the process, k  5, to include as much as the process measurements being assessed. The accessibility of loss and risk using IBLF constructed from the standard beta distribution can be generalized (Appendix A). Whenever the process is to be assessed by IBLF having a finite range (beta, uniform or any truncated distributions) or infinite range (normal, gamma, Weibull distributions, etc.) the loss and risk can still be evaluated. In general, the scale invariant nature of the IBLF and its associated risk function under linear transformation holds for any distribution having a unique maximum. There are some limitations of this IBLF when the unique maximum conditions do not hold. For example, taking a ¼ 1 and b ¼ 1 with any target value T, the loss will be zero over the range (0, 1) when standard beta is the underlying distribution. It is unrealistic to have zero loss between the two specification limits.

Appendix A If Y has a generalized beta distribution with parameters a > 0, b > 0 and ranging from p to q (with p  q), then we can transform it to a standard beta distribution that possesses the same loss function as X. The pdf of Y is:     1 y  p a1 q  y b1 gðyÞ ¼ ; p < y < q: Bða; bÞðq  pÞ q  p qp Let Y ¼ ðq  pÞX þ p and then X ¼ with

  dy  jJ j ¼   ¼ ðq  pÞ: dx

Similarly, the mode of Y can be obtained through differentiation and found to be y¼

ðq  pÞða  1Þ þ p; aþb2

and let it equal to T 0 , the target with respect to Y. Note that, 1.

T 0 p qp

a1 ¼ aþb2 ¼ T ; the target value in X ;

2. b  1 ¼ 1T T ða  1Þ; and 

0 ð1T Þ=T1a n o1a ð1T Þ=T T 0 p qT 3. ¼ T ð1  T Þ ¼ C: qp qp So,

Acknowledgement

Y p ; qp

    1 y  p a1 q  y b1 ; pðy; T Þ ¼ Bða; bÞðq  pÞ q  p qp 0

We would like to express our appreciation to the two anonymous referees whose suggestions greatly improved the manuscript. Their comments were insightful and helpful, thank you.

and correspondingly m0 , the supremum of pðy; T 0 Þ, is  0    1 T  p a1 q  T 0 b1 m0 ¼ Bða; bÞðq  pÞ q  p qp 1 C 1 : ¼ Bða; bÞðq  pÞ

References

Hence the loss inversion ratio     pðy; T 0 Þ y  p a1 q  y ðð1T Þ=T Þða1Þ ¼C m0 qp qp n oa1 ¼ C xð1  xÞð1T Þ=T ;

Spiring, F.A. (1993) The reflected normal loss function. Canadian Journal of Statistics, 3, 321–330. Spiring, F.A. and Yeung, A.S. (1998) A general class of loss functions with industrial applications. Journal of Quality Technology, 30, 152–162. Sun, F., Laramee, J. and Ramberg, J. (1996) On Spiring’s inverted normal loss function. Canadian Journal of Statistics, 2, 241–249. Taguchi, G. (1986) Introduction to Quality Engineering: Designing Quality into Products and Processes, Kraus, White Plains, NY. Wolfram, S. (1991) Mathematica: A System for Doing Mathematics by Computers, 2nd edn, Addison Wesley.

ðwhich is same as Equationð1ÞÞ ¼

pðx; T Þ ; m ) Lðx; T Þ ¼ Lðy; T 0 Þ:

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Leung and Spiring

Appendix B Follows from the proof of Properties 1 and 2, the pdf of X is f ðxÞ ¼ gðyÞjJ j ¼ g½xðq  pÞ þ pðq  pÞ; 1 ¼ xa1 ð1  xÞb1 ðq  pÞ; Bða; bÞðq  pÞ 1 xa1 ð1  xÞb1 ; 0 < x < 1: ¼ Bða; bÞ

As another example to show the ease of evaluating IBLF. Let’s consider if X U ðaR ; bR Þ, where 0 < aR < bR , then the associated risk function is:

E½LðX ; T Þ ¼

aR

E½LðY ; T 0 Þ ¼

¼

¼

¼

aR

9 = xa1 ð1  xÞðð1T Þ=T Þða1Þ dx ; ;

Lðx; T ÞgðyÞdy;

9 8 pffiffiffiffi arcsin bR > > > > Z = < 2C 2lþ1 2nþ1 E½LðX ; T Þ ¼ K 1  sin h cos hdh ; > > bR  aR > > pffiffiffiffi ; : arcsin aR

Lðx; T Þg½xðq  pÞ þ pðq  pÞdx;

0

Z1

ZbR

let l ¼ a  1, n ¼ ðð1  T Þ=T Þða  1Þ, both be positive integers, and let x ¼ sin2 h, dx ¼ 2 sin h cos h dh

p

Z1

8