Generalized Trapezoidal Distributions

J. René van Dorp, Samuel Kotz

Department of Engineering Management & Systems Engineering, The George Washington University, 1776

G Street, N.W., Washington D.C. 20052 (e-mail: [email protected]).

Submitted April 24, 2002; Revised, March 2003

Abstract. We present a construction and basic properties of a class of continuous distributions of an arbitrary form defined on a compact (bounded) set by concatenating in a continuous manner three probability density functions with bounded support using a modified mixture technique. These three distributions may represent growth, stability and decline stages of a physical or mental phenomenon.

Keywords: Mixtures of Distributions, Risk AnalysisÄ Applied Physics

1. INTRODUCTION Trapezoidal distributions have been advocated in risk analysis problems by Pouliquen (1970) and more recently by Powell and Wilson (1997). They have also found application as membership functions in fuzzy set theory (see, e.g. Chen and Hwang (1992)). Our interest in trapezoidal distributions and their modifications stems mainly from the conviction that many physical processes in nature and human body and mind (over time) reflect the form of the trapezoidal distribution. In this context, trapezoidal distributions have been used in the screening and detection of cancer (see, e.g. Flehinger and Kimmel, (1987) and Brown (1999)). Trapezoidal distributions seem to be appropriate for modeling the duration and the form of a phenomenon which may be represented by three stages. The first stage can be

To appear in Metrika, Vol. 58, Issue 1, July 2003 2 viewed as a growth-stage, the second corresponds to a relative stability and the third represents a decline (decay). These distributions however are restricted since the growth and decay (in the first and third stages) are limited in the trapezoidal case to linear forms and the second stage represents complete stability rather than a possible mild incline or decline. The trapezoidal probability density function is of the form ¿ Â ?çüB+ +éB, Â ,+ ? ,éB- 0µBl+Ä,Ä-Ä.¶æÀ µ"¶ Â ?çü.B -éB. Â .- Á ! /6=/A2/

fX(x) 2 d+c-b-a

ab c dx

Figure 1. Probability Density Function of a Trapezoidal Distribution

Another domain for applications of the trapezoidal distribution is the applied physics arena (see, e.g. Davis and Sorenson (1969), Nakao and Iwaki (2000), Sentenac et al. (2000), Straaijer and De Jager (2000) ). Specifically, i n the context of nuclear engineering, uniform and trapezoidal distribution have been assumed as models for observed axial

To appear in Metrika, Vol. 58, Issue 1, July 2003 3 distributions for burnup credit calculations (see, Wagner and DeHart (2000) and Neuber (2000) for a comprehensive description). These distributions are important to burnup credit criticality safety analyses for pressurized-water-reactor (PWR) fuel. Figure 2Ä adapted from Wagner and DeHart (2000)Ä depicts the actual data and axial burnup distributions for two profiles of normalized burnup versus percent axial height (using interpolation between observed data points). The uniform distribution has been shown to be only conservative for low burnupsÄ not when burnup increases (see, Wagner and DeHart (2000)). The use of trapezoidal distributions tend to result in conservative analyses (see, Neuber (2000)). The modeling of axial burnup distributions has been recognized as an important and timely research area in nuclear engineering (See, Parks et al. (2000)).

Normalized Burnup 1.4 Axial Height % Profile 1 Profile 2 2.78% 0.652 0.649 1.2 8.33% 0.967 1.044 13.89% 1.074 1.208 1 19.44% 1.103 1.215 25.00% 1.108 1.214 30.56% 1.106 1.208 0.8 36.11% 1.102 1.197 41.69% 1.097 1.189 0.6 47.22% 1.094 1.188

57.80% 1.094 1.192 Normalized Burnup 0.4 58.33% 1.095 1.195 63.89% 1.096 1.190 0.2 69.44% 1.095 1.156 75.00% 1.086 1.022 0 80.56% 1.059 0.756 0.00% 20.00% 40.00% 60.00% 80.00% 100.00% 86.11% 0.971 0.614 Axial Height(%) 91.67% 0.738 0.481 97.22% 0.462 0.284 Profile 1 Profile 2

Figure 2. Two Typical Profiles of Observed Axial Distributions in PWR

In the case of the distribution given by µ"¶ both the growth and decay stages are linear and the density at ,- and is

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" 0\\ µ,¶ æ 0 µ-¶ #µ. þ - , +¶, µ#¶ where +é,é-é. . We shall strive for a continuous generalization of the trapezoidal distribution where the growth and decay may exhibit a nonlinear convex or concave behavior and the densities 0µ,¶\\ and 0µ-¶ do not necessarily have to take the same value. Rather, a boundary ratio parameter !! is introduced such that 0µ,¶æ\\ 0µ-¶ . Generalized trapezoidal distributions herein inherit the four basic trapezoidal parameters

+Ä ,Ä - and . and need, for complete specification, two additional parameters 8"$ and 8 specifying the growth rate and decay rate in the first and third stage of the distribution, in addition to the boundary ratio parameter ! . An advantage of the generalized trapezoidal distribution is in its flexibility which allows us inter alia to appropriately mimic the great variety of the growth and decay behaviors. In Section 2, the functional form of the generalized trapezoidal distribution is derived to be

0µBl\ +Ä,Ä-Ä.Ä8Ä8Ä"$! ¶æ

¿ 8" Â #88! " Â "$ çüB+ +éB, Â #!! µ,+¶8$"$" þµ þ"¶µ-,¶8 8 þ#µ.-¶8 ,+ Â #8"$ 8 èýµ! "¶-B þ" ,éB- À #!! µ,+¶8$"$" þµ þ"¶µ-,¶8 8 þ#µ.-¶8 -, µ$¶ Â 8"$ Â #8"$ 8 çü.B -éB. Â #!! µ,+¶8$"$" þµ þ"¶µ-,¶8 8 þ#µ.-¶8 .- Á ! /6=/A2/

¿ $ !1 0 µB¶ +éB. 3\3 0µB¶æ\ À 3æ" µ%¶ Á ! /6=/A2/

$ ! where 1133æ"Ä ë!, with 3æ"

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8" 8" B+ " 0\" µBl+Ä,Ä8¶æçüçü, +éB,, 8 " ë!Ä µ&¶ " ,+ ,+

# 0\ µBl,Ä-Ä!!!! ¶æî µ" ¶Bþ -, Ä,éBé-Ä ë!Ä µ'¶ # µ! þ "¶() - , #

8" 8.B$ $ 0\$ µBl-Ä.Ä8¶æçüçü Ä-éB.Ä8 $ ë!. µ(¶ $ .- .-

Note that, the density function in the second stage is restricted to a linear form such that 0 µ,l,Ä -Ä!! ¶ æ 0 µ-l,Ä -Ä ! ¶ ! ! " ! ë " µ'¶ \\## . For ( ) the density in exhibits an inclining (declining) behavior. For ! æ" , µ(¶ reduces to a uniform density on ·,Ä-¸ . Figure 3 depicts two members in the generalized trapezoidal family that closely follow the axial distribution profiles in Figure 2. From Figure 3 it can be concluded that the density function of the generalized trapezoidal distribution (cf. µ$¶ ) may well be geared towards modeling axial distribution profiles. Note especially Case B, where the decline in the central part is closely tracked. Applications to reliability and risk analysis may also become more appropriate by replacing the linear parts with a power function.

1.4 A 1.4 B

1.2 1.2

1 1

0.8 0.8

0.6 0.6

0.4 0.4 Normalized Burnup Normalized Burnup 0.2 0.2

0 0 0.00% 20.00% 40.00% 60.00% 80.00% 100.00% 0.00% 20.00% 40.00% 60.00% 80.00% 100.00% Axial Height (%) Axial Height (%) Gen. Trapezoidal Profile 1 Gen. Trapezoidal Profile 2

Figure 3. Generalized Trapezoidal approximation of Axial Distributions depicted

in Figure 2; A: + æ !Ä , æ !Ã"&Ä - æ !Ã)Ä . æ "Ä 8"$ æ "Ã#&Ä 8 æ "Ã%& , ! æ "Å

B: + æ !Ä , æ !Ã"%Ä - æ !Ã'*Ä . æ "Ä 8"$ æ "Ã$&Ä 8 æ "Ã(&, ! æ "Ã!%.

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Some additional examples of generalized trapezoidal distributions will be presented in Section 2. In Section 3 the "mixing" behavior in µ%¶ will be studied for some limiting cases. In Section 4 we shall briefly discuss some basic properties associated with µ$¶ . Concluding remarks are provided in Section 5.

2. CONSTRUCTION OF PROBABILITY DENSITY FUNCTION Our approach towards constructing the desired distribution requires to specify: µ"¶ the ends and beginnings of the three stages µ+Ä,Ä-Ä.¶Ä µ#¶ the growth behavior of the first stage (parameter 8" ¶Ä µ$¶ the decay behavior of the third stage µ parameter 8$ ¶ and µ%¶ the relative likelihood of capabilities at the end of the growth stage·+Ä ,¸ and at the beginning of the decay stage·-Ä .¸ , namely the boundary ratio parameter

! æ 0\\ µ,¶³0 µ-¶. µ)¶

To allow for nonlinear growth and decay the probability density functions µ&¶ and µ(¶ are chosen for \\µ%¶"$ and in , respectively Ã The density function at the second stage will be restricted to the linear form given by µ'¶ satisfying (as previously noted) 0 µ,l,Ä -Ä!! ¶ æ 0 µ-l,Ä -Ä ! ¶Ã µ*¶ \\##

The main challenge in the construction is to select the remaining mixing probabilities

111"#$ÄÄ in µ%¶ so that the overall density function in µ%¶ be continuous. This turns out to be a nontrivial problem.

Proposition: The probability density function given by µ$¶ follows from expressions µ%¶Ä µ&¶Ä µ'¶Ä µ(¶and µ)¶ utilizing mixture probabilities

¿ #µ,+¶8! $ Â 1" æ Â #!! µ,+¶8$"$" þµ þ"¶µ-,¶8 8 þ#µ.-¶8 µ! þ"¶µ-,¶8"$ 8 À 1# æ #!! µ,+¶8 þµ þ"¶µ-,¶8 8 þ#µ.-¶8 µ"!¶ Â $"$" #µ.-¶8" Á 1$ æ #µ,+¶8þµþ"¶!!$ µ-,¶8"$ 8 þ#µ.-¶8 ",

To appear in Metrika, Vol. 58, Issue 1, July 2003 7 where +,-. , 8"# ë!Ä8 ë!Ä! ë! and the probability density function given by µ$¶ is continuous.

Proof: Utilizing µ%¶ , µ&¶ , µ'¶ and µ(¶ the density function of the proposed generalized trapezoidal distribution given by µ%¶ can be rewritten as ¿ 1 0 µBl+Ä,Ä8¶ +éB, Â "\" " 1!0 µBl,Ä-Ä ¶ ,éB- 0 µBl@ ¶ æÀ #\# µ""¶ \ 1 0 µBl-Ä.Ä8¶ -éB. Â $\$ $ Á ! /6=/A2/

$ +,-.Ä"1!3"$ æ"Ä8 ë!Ä8 ë!Ä ë!. µ"#¶ 3æ"

It will be convenient to write the mixture weights 13 , 3æ"Ä#Ä$Ä in the form

1"1"#æ :Ä æ µ" "1" ¶Ä $ æ µ" :¶ µ"$¶ where 0"1 "Ä 0 :" . This implies that 3 ë!Ä3æ"ÄÆÄ$ and $ "1"3 æ : þµ" "" ¶þ µ":¶æ". µ"%¶ 3æ"

From µ)¶ , utilizing µ&¶ , µ(¶ , µ""¶ and µ"$¶ , we have 0 µ,l@ ¶" :0 µ,l+Ä ,Ä 8 ¶ :µ. -¶8 ! æ\ æ\"" æ " µ"&¶ 0þ µ-l@ ¶" µ" :¶0 µ-l-Ä .Ä 8 ¶ µ" :¶µ, +¶8 , \ \$$ $

þ where 0 µ,l@@@@ ¶ æ 637 0\\ µBl ¶ and 0 µ-l ¶ æ 637 0 µBl ¶ , yielding \ Bª,\ B«-

µ, +¶8$! :æ. µ"'¶ µ. -¶8"$ þ µ, +¶8 !

Observe that : does not depend on " . Also the stipulations +,-. ,

8ë!Ä8ë!"$ and ! ë! imply !:"Ã

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Continuity of µ""¶ at , will follow from the requirement that þ 0\ µ,||@@ ¶ æ 0\ µ, ¶Ä µ"(¶ implying with µ"$¶and µ""¶ that ""!:0 µ,l+Ä,Ä8 ¶æµ" ¶0 µ,l,Ä-Ä ¶ µ" ¶ \""# \ .8

Utilizing µ&¶Ä µ'¶Ä µ")¶and µ"'¶ we obtain #µ. -¶8 þ #! µ, +¶8 " æ"$ Ã µ"*¶ #µ. -¶8""$$ þ µ!! þ "¶µ- ,¶8 8 þ # µ, +¶8

From +,-.Ä8ë!Ä8ë!"$ and!" ë! it follows that ! "Ã The choice of

" in µ"*¶ assures continuity of 0\ µ ÷ l@¶ (cf. µ""¶ ) at , . Utilizing µ")¶ , µ*¶ and µ"&¶ , it follows that ""!µ" :¶0 µ-l-Ä .Ä 8 ¶ æ µ" ¶0 µ-l,Ä -Ä ¶Ã µ#!¶ \$$# \

The continuity of 0\ µ ÷ l@ ¶ (cf. µ""¶ ) at - is implied by µ"$¶ and µ#!¶ . Substituting µ"'¶ and µ"*¶ into µ"$¶ we arrive at

¿ #µ,+¶8! $ Â 1" æ Â #!! µ,+¶8$"$" þµ þ"¶µ-,¶8 8 þ#µ.-¶8 µ! þ"¶µ-,¶8"$ 8 À 1# æ #!! µ,+¶8 þµ þ"¶µ-,¶8 8 þ#µ.-¶8 µ#"¶ Â $"$" #µ.-¶8" Á 1$ æ #µ,+¶8þµþ"¶!!$ µ-,¶8"$ 8 þ#µ.-¶8 ".

Finally, substitution of µ&¶ , µ'¶ , µ(¶ and µ#"¶ into µ""¶ , yields µ$¶Ã

The conditions in µ&¶ and µ(¶ stipulated 8"# ë ! and 8 ë ! . To adhere to the truly

"trapezoidal" shape one may restrict 8ë""# and 8ë" in the first and third stages. In case !8"$ "Ä!8 " the first stage reflects decay and the third expresses growth of the density 0µBl¶\ @ given by µ$¶ resulting in a "bathtub" shape rather than a trapezoidal shape for the combined density. Figure 4 displays different shapes of generalized trapezoidal distributions.

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1.5 1.2 1 1 0.8 < < < < = = 1 n1 2, 1 n 3 2 n1 2, n 3 2 0.6 α = 0.5 0 < α < 1 0.4 1 0.2 0 0 0.00 0.20 0.40 0.60 0.80 1.00 0.00 0.20 0.40 0.60 0.80 1.00

2 1.5

1.5 1 1 < < > > > 1 n1 2, n 3 2 n1 2, n 3 2 0.5 < α < 0.5 α > 1 0 1 0 0 0.00 0.20 0.40 0.60 0.80 1.00 0.00 0.20 0.40 0.60 0.80 1.00

1.4 2.5 1.2 2 0 < n < 1, 0 < n < 1 1 1 3 1.5 α > 1 0.8 n > 2, 1 < n < 2 0.6 1 3 1 0.4 α = 1 0.5 0.2 0 0 0.00 0.20 0.40 0.60 0.80 1.00 0.00 0.20 0.40 0.60 0.80 1.00

2.5 2.5 < < < < > < < 2 1 n1 2, 0 n 3 1 2 n1 2, 0 n 3 1 1.5 0 < α < 1 1.5 α = 1 1 1 0.5 0.5 0 0 0.00 0.20 0.40 0.60 0.80 1.00 0.00 0.20 0.40 0.60 0.80 1.00

3 2.5 2.5 < < < < 0 < n < 1, n > 2 0 n1 1, 1 n 3 2 2 1 3 2 < α < α > 1 1.5 0 1 1.5 1 1 0.5 0.5 0 0 0.00 0.20 0.40 0.60 0.80 1.00 0.00 0.20 0.40 0.60 0.80 1.00

Figure 4. Generalized Trapezoidal DistributionsÃ

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The graphs in Figure 4 alternate between the three cases !!! "Ä æ" and ! ë"Ã

Substituting 8æ8æ#"$ and ! æ" into µ$¶ we arrive at the trapezoidal distribution given by µ"¶ . 0÷ We note in passing that \# ( ) can be taken to be a conditional Two Sided Power density (see, Van Dorp and Kotz (2002)) on ·+Ä .¸ truncated to ·,Ä -¸ (rather than the linear form in µ'¶ ), which results in an extension of the trapezoidal distribution (cf. µ"¶ ) permitting mild oscillation in the central stage.

3. MIXING BEHAVIOR Some insight about the mixing behavior for generalized trapezoidal distributions in µ$¶ can be gained by studying limiting behavior of the mixing probabilities in µ"!¶ . From µ"*¶ we have " æ µ" þ K¶" Ä where µ! þ "¶µ- ,¶ K æ. µ##¶ #µ.-¶þ #! µ,+¶ 88$"

Since +,-.Ä8ë!Ä8ë!"$ and !" ë! we have Kë! and the largest (least) corresponds to least (greatest) K . As 8"$ ©_ and 8 ©_ , K©_ and therefore " «!

(limiting "least" case¶ . Hence, from 1"# æ " (cf. µ"$¶ ) it follows that no probability mass is attributed to the first and last stages in the limit when 8©_"$ and 8©_ and µ$¶ 0 µBl,Ä -Ä!" ¶ µ'¶ 8 « ! 8 « !Ä K « ! ª converges to \"$# (cf. ). As and and 1 (limiting

"greatest" case). Hence, from 1"# æ " µ cf. µ"$¶ ) it follows that all the probability mass is attributed to the first and last stages in the limit as 8"$ « ! and 8 « ! µ cf. µ&¶ and µ(¶ ).

8" µ#"¶ 8"$ « ! 8 « ! æ V From it follows that letting and while keeping 8$ (constant) we have #µ,+¶!V #µ.-¶ 11© Ä © µ#$¶ 1 #!V!V µ,+¶þ#µ.-¶$ # µ,+¶þ#µ.-¶ .

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8 «! 8 «!Ä 0 µBl+Ä,Ä8 ¶ It is easy to verify that as "$ and the density \"" converges to a " + 0 µBl-Ä.Ä8 ¶ single point mass of at and the density \$$ converges to a single point mass of ". at . Thus, µ$¶ converges to a transformed Bernoulli distribution assigning the limiting probability 11"$ in µ#$¶ to + and limiting probability in µ#$¶ to .Ã

Letting 8"$ « ! and keeping 8 fixed, it follows from µ#"¶ that in this case 1 " ª "Ä

11#$«! and «!Ã Hence, all the probability mass is attributed to the first stage Ã It is easy 8 «! 0 µBl+Ä,Ä8 ¶ to verify that when "\" the density " converges to a single point mass of

"+ at . Vice versa, letting 8©_"$" and keeping 8 fixed, we have 1 «! . Here no probability mass is attributed to the first stage and

µ! þ "¶µ- ,¶8$ #µ. -¶ 11#$© Ä © Ã µ#%¶ µ!! þ "¶µ- ,¶8$$ þ #µ. -¶ µ þ "¶µ- ,¶8 þ #µ. -¶

µ$¶ 0 µBl! Ä ,Ä -¶ 0 µBl-Ä .Ä 8 ¶ Consequently, reduces to a mixture of \\$#$ and assigning the limiting probability 11# in µ#%¶ to the first density and the limiting probability $ in µ#%¶ to the second densityÃ8«!8 Analogous conclusions can be drawn letting $" , keeping fixed.

4. BASIC PROPERTIES In the sections below we shall briefly investigate the cumulative distribution and the moments of the generalized trapezoidal type distributions.

4.1. Cumulative Distribution Function

The cdf associated with \ 0\ µBl@ ¶ in µ$¶ can be derived using µ""¶Ä µ"!¶Ä µ&¶ , µ'¶ and µ(¶ yielding

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¿ Â ! B+ Â 8" Â #µ,+¶8! $ çüB+ +éB, Â #!! µ,+¶8$"$" þµ þ"¶µ-,¶8 8 þ#µ.-¶8 ,+ µ! "¶ µ#-,B¶ #! µ,+¶8$"$ þ#µB,¶8 8è "þ ý JµBl¶æ\ @ À #µ-,¶ ,éB-Ä µ#&¶ Â #!! µ,+¶8$"$" þµ þ"¶µ-,¶8 8 þ#µ.-¶8 Â 8$ Â "#µ.-¶8" çü.B -éB. Â #!! µ,+¶8$"$" þµ þ"¶µ-,¶8 8 þ#µ.-¶8 .- Á " Bë.

Setting 8æ8æ#"$ and ! æ"in µ#&¶yields ¿ ! B+ Â Â # Â µ,+¶ B+ +éB, Â .þ-,+çü ,+ J µBl+Ä,Ä-Ä.¶æÀ µ,+¶þ#µB,¶ ,éB- Ä µ#'¶ \ Â .þ-,+ Â # Â "µ.-¶ çü.B -éB. Â .þ-,+ .- Á " Bë. which is recognized as the cdf of the standard trapezoidal density given by µ"¶Ã

4.2. Moments

Utilizing µ%¶ and µ"$¶Ä the 5 -th moment of \ 0\ µBl@ ¶ (cf. µ$¶ ) may be derived as

55 5 5 I·\ l@ ¸ æ""!" :I·\"#$ l+Ä ,Ä 8"$ ¸ þ µ" ¶I·\ l,Ä -Ä ¸ þ µ" :¶I·\ l-Ä .Ä 8 ¸Äµ#(¶ where " , : are given by µ"*¶ and µ"'¶ , respectively Ã The pdf’s of \"# Ä \ and \ $ are defined in µ&¶Ä µ'¶ and µ(¶ , respectively. Numerical calculations of 5 -th moment I·\5 l@ ¸given by µ#(¶ are quite straightforward employing the current advances in computer technology. Deriving a closed form for the expression of I·\5 l@ ¸ for

\0\ µBl@ ¶ (cf. µ$¶ ) in its general form, although tedious, does not present intrinsic difficulties. We shall conclude by providing closed form expressions for the first and second moments of a generalized trapezoidal variable \ . Fromµ&¶ and µ(¶ we obtain that

+þ8,"$ 8-þ. I·\l+Ä,Ä8¸æ"" ÄI·\l-Ä.Ä8¸æ $$ Ã µ#)¶ 8þ""$ 8þ"

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Utilizing µ'¶Ä yields # µ!! "¶µ-$$ , ¶þµ -,¶µ- ## , ¶ I·\ l,Ä -Ä! ¸ æ$ µ#*¶ # µ-,¶µ# ! þ"¶

Hence from µ#(¶ (setting 5 æ " ) Ä µ#)¶Ä µ#*¶Ä µ"'¶ and µ"*¶ we have I·\l@ ¸ æ µ$!¶

# $$ ## +þ8,"$µ!! "¶µ- , ¶µ -,¶µ- , ¶ 8-þ.$ #! µ,+¶8$"$çü8þ" 8 8 ç µ-,¶ ü þ#µ.-¶8 " çü 8þ" " $ Ã #!! µ, +¶8$"$" þ µ þ "¶µ- ,¶8 8 þ #µ. -¶8

Analogously, from µ&¶Ä µ'¶ and µ(¶ we obtain

## # #+þ#8+,þ8µ8þ"¶,""" I·\ l+Ä,Ä8" ¸ æ " µ8"" þ#¶µ8 þ"¶ "# µ!! "¶µ-%% , ¶þ µ -,¶µ- $$ , ¶ # #$ µ$"¶ I·\# l,Ä-Ä! ¸ æ µ-,¶# µ! þ"¶ # # # #.+ #8$$ -.þ8 µ8$ þ"¶- I·\ l-Ä.Ä8$ ¸ æ Ã $ µ8$$ þ#¶µ8 þ"¶

Using µ$"¶ , the second moment I·\# l@ ¸ now follows from µ#(¶ (setting 5 æ # ), µ"'¶ and µ"*¶ to be

## I·\# l@ ¸ æ#! µ,+¶8$"""çü #+ þ#8 +,þ8 µ8 þ"¶, #!! µ,+¶8$"$""" þµ þ"¶µ-,¶8 8 þ#µ.-¶8 µ8 þ#¶µ8 þ"¶ " µ! "¶µ-%% , ¶# µ! -,¶µ- $$ , ¶ 88"$ç # $ ü þ µ$#¶ #!! µ,+¶8$"$" þµ þ"¶µ-,¶8 8 þ#µ.-¶8 µ-,¶ ## #µ.-¶8"$$$çü 8 µ8 þ"¶- þ#8 -.þ#. Ã #!! µ,+¶8$"$"$$ þµ þ"¶µ-,¶8 8 þ#µ.-¶8 µ8 þ#¶µ8 þ"¶

The variance of a generalized trapezoidal variable \ may be calculated utilizing µ$"¶ and

µ$#¶. Setting 8"$ æ 8 æ # and ! æ " in µ$"¶ and in µ$#¶ , we have the elegant formulas µ, +¶µ+ þ #,¶ $µ,## - ¶ þ µ. -¶µ - þ .¶ I·\l+Ä ,Ä -Ä .¸ æ2 Ä µ$$¶ $µ. þ - , +¶

µ, +¶ " " I·\### l+Ä ,Ä -Ä .¸ æçü µ+ þ ,¶ þ , þ µ$%¶ µ.þ-,+¶ '$ "# µ.-¶"" çüµ-$$ , ¶ þ ç - # þ µ- þ .¶ # ü µ.þ-,+¶ $ µ.þ-,+¶ $ ' , for the first and second moment of the standard trapezoidal density given by µ"¶ .

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5. CONCLUDING REMARKS In the course of the construction some interesting features emerged which may be worthy of specific mention. Firstly the structure of these distributions - although formally a mixture of three components - differs from the commonly encountered mixtures in two aspects: (1) the mixing parameters are of a special form (a product of two quantities (cf. µ"$¶) each performing a function needed to properly link the three components in (cf. µ""¶) and (2) the components represent different distributions each capable of taking a variety of forms. Next, while classical continuous distributions are characterized by the property that continuity is generated by means of a mathematical function that forces a special form of the distribution, here continuity is generated by linking appropriately the three relevant parts of the distribution rendering an additional flexibility. We have attempted to demonstrate a method of constructing versatile and flexible family of continuous distributions on a compact set. The procedure depends on the values of the parameters of the constituent distributions and provides an example of a new form of a mixture consisting of nonlinear components. The family has transparent physical interpretation and potential applications in engineering, behavioral and medical sciences

6. ACKNOWLEDGMENT

The authors are indebted to the Editor of Metrika and the referee for their most valuable comments and suggestions which improved both the content and streamlined the presentation.

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