Using the Average of the Extreme Values of a Triangular Distribution for a Transformation, and Its Approximant via the Continuous Uniform Distribution

Transcription

1 British Journal of Mathematis & Computer Siene 4(4):., 014 ISSN: SCIENCEDOMAIN international Using the Average of the Extreme Values of a Triangular Distribution for a Transformation, and Its Approximant via the Continuous Uniform Distribution H. I. Okagbue 1, S. O. Edeki 1, A. A. Opanuga 1, P. E. Oguntunde 1 and M. E. Adeosun 1 Department of Mathematis, College of Siene & Tehnology, Covenant University, Otta, Nigeria. Department of Mathematis and Statistis, Osun State College of Tehnology, Esa-Oke, Nigeria. Artile Information DOI: /BJMCS/014/199 Editor(s): 1. Reviewers: Peer review History: Reeived: 6 June 014 Aepted: 18 August 014 Original Researh Artile Published: 09 Otober 014 Abstrat This paper introdues a new probability distribution referred to as a transformed triangular distribution (TTD) by using the average of the extreme values (minimum and maximum) of the triangular distribution. The TTD is being approximated by the ontinuous uniform distribution. The basi moments of the TTD and those of the ontinuous uniform distribution are ompared respetively, and a relationship established. This an be used in modeling and simulation. Keywords: Moments, uniform distribution, triangular distribution, transformed distribution, ontinuous random variable. Mathematial Subjet Classifiation (010): 05A10, 81S0, 81S30, 33B0 Corresponding author: soedeki@yahoo.om;

2 British Journal of Mathematis & Computer Siene 4(4),.., Introdution Triangular distribution is a ontinuous distribution with a fixed minimum, fixed maximum and most likely value to our (mode). The most likely value lies between the minimum and maximum values, forming a triangular-shaped distribution whih shows that values near the minimum and maximum are less likely to our [1]. The minimum and the maximum values are alled the extreme values. The values an either be symmetrial (the mod e = = median ) or asymmetrial []. The distribution is widely used to approximate Beta distribution [3]. Some of the earliest known written work on the triangular distribution are traeable to the work of [4,5], even though it was Simpson that first mentioned the distribution in his papers [6]. Nevertheless, many authors and researhers have worked on the triangular distribution; some of them are either on the statistial, probabilisti nature of the distribution or in the area of simulation and modeling. Some of the ontributions inlude the harateristis of the distribution [7,8], the produt of two identially independent distributed triangular variables [9], on the extension of the triangular distribution and the onvolution [10,11], appliations in projet evaluation and review tehnique PERT [1-16], non-smooth sailing with respet to asymptoti distributions [17,18], Monte Carlo Simulation [19-], the properties of bivariate triangular distribution [3-4], on the negative binomial-triangular distribution [5], advaned simulation and risk modeling [6-8] ombination of triangular and exponential distributions [9], the sum of two triangular distributions [30], disrete nature of triangular distribution and non-parametri estimation for probability mass funtion [31]. The remaining part of the paper is strutured as follows: setion deals with the basi onepts and properties of the onerned probability distributions, setion 3 is on methodology and proedures towards the transformation of the triangular distribution and disussion of results; while in setion 4, a onluding remark is made. The Basi Conepts of the Probability Distributions In this setion, we present both the properties of the triangular and the ontinuous uniform distributions to be used in the later part of the work..1 Some Moments and Properties of the Triangular Distribution (TD) Suppose is a random variable with parameters and a support, a x b a, b and a : a (, ) b : a b : a < suh that 3498

3 British Journal of Mathematis & Computer Siene 4(4),.., 014 then, is said to follow a triangular distribution T ( a, b,) with the following properties: Probability density funtion (pdf) f ( x ) of given as: 0, for x < a ( x a), for a x b ( a)( b a) f ( x) = ( x), for a x ( a)( b) 0, for x > (1) Cumulative Density Funtion (CDF) : 0, for x < a ( x a), for a x b ( a)( b a) F( x) = ( x) 1, for b < x ( a)( b) 1, for < x () Mean: a + b + E( ) = (3) 3 Median: Most likely value Median ( a)( b a) a + for b = ( a)( b) for b (4) Mode = b (5) Variane: 3499

4 British Journal of Mathematis & Computer Siene 4(4),.., 014 a + b + a ab b Var( ) = (6) 18 Skewness: Skewness = ( b)( a b)( a + b) 3 5( + b a ab b) (7) Kurtosis: = 3 / 5 (8) Kurtosis. Some Moments and Properties of the Continuous Uniform Distribution A random variable over the interval I = [ a, b] is a ontinuous uniform distribution if it is equally likely to assume any value in I. Let f ( y) and F( y ) be the pdf and CDF of respetively, then: 1, a y b f ( y) = (b a) (9) 0, otherwise 0, for y < a ( y a) F( y) =, for a < y b ( b a) 1, for b (10) Remark.1: The following an easily be omputed. Mean: b + a = (11) Mode (an be any value in I ) so we hoose: Median mode b + a = (1) 3500

5 British Journal of Mathematis & Computer Siene 4(4),.., 014 Variane median b + a = (13) ( b a) Var( ) = (14) 1 Coeffiient of skewness = 0 (15) skewness 3 Methodology and the Modifiation of the Triangular Distribution 3.1 The Transformation of the Triangular Distribution (TTD) We replae the most likely value b, with the average of the minimum and maximum of the Triangular distribution in order to modify the Triangular distribution, thus; b = (16) The resulting distribution will heneforth be referred to as the Transformed Triangular Distribution (TTD). 3. The Resulting Distribution TTD and Its Properties Suppose is the random variable assoiated with the TTD, f ( x ) and F ( x ) as the orresponding pdf and CDF respetively, then by using (9), one an easily obtain the following: 4( x a), a x ( a) 4( x) ( a) 0, otherwise f ( x) =, x (17) and it is easy to show that: 3501

6 British Journal of Mathematis & Computer Siene 4(4),.., 014 0, for x < a ( x a), for a x ( a) F ( x) = ( x) 1, for x < ( a) 1, for x (18) Remark 3.1 validation of the probability density funtion of f ( x ) To validate the pdf of the TTD, we need to show that: Proof: By definition, f ( x) dx = 1 (19) a 4( x a) 4( x) f ( x) dx = dx + dx a a ( a) ( a) where 4 = ( x a) dx ( a) dx + ( a) a = 4 { A B} ( a) + (0) Thus, Sine ( a ) x ( a) x A = ax = & B = x = 8 8 a 4 f ( x) dx = A + B ( a) a = 1 { } (1) () ( a ) ( ) ( ) A + B = & a a, a (3) 4 350

7 British Journal of Mathematis & Computer Siene 4(4),.., 014 Showing that f ( x ) is indeed a valid pdf. For the rest of the moments and properties we shall often refer to (17), as suh (16) in (3) gives: Mean: Thus, showing that: a + + = 3 b + a = (4) (5) The Median of TTD using the extreme values: Substitute equation (16) in (4) gives: showing that: median = a + ( a)( a) ( a)( a) = a + 4 = median (6) median (7) median Similarly ( a)( ( ) Median = ( a)( a) = 4 = showing the same result in (7). 3503

8 British Journal of Mathematis & Computer Siene 4(4),.., 014 The Mode of TTD ( an be any value in I ): In this ase of mode, hoose b suh that: mode = b = mode (8) showing that: mod e (9) mod e The variane of the TTD: Substitute (16) in (6) gives: showing that: Var ( ) a + ( ) + a a( ) ( ) = 18 ( a) = 4 Var( ) (30) Var ( ) Var( ) (31) The skewness of the TTD: From (16) b = = b = (3) Therefore, substituting (16) and (3) in (7) gives: Skewness ( b b) a a + = 3 ( ) ( ) ( ) a a a a a 4 = 0 = skewness (33) 3504

9 British Journal of Mathematis & Computer Siene 4(4),.., Conlusion In this paper, we have showed that when the mode of a triangular distribution is the average of the extreme values, then the resulting distribution referred as transformed triangular distribution (TTD) is a probability funtion that an be reasonably approximated, simulated and modeled by the ontinuous uniform distribution. The limit theorem takes are of the behavior of the distribution at large sample. Aknowledgements We would like to express sinere thanks to the anonymous reviewer(s) and referee(s) for their onstrutive omments and valuable suggestions towards the improvement of the paper. Competing Interests Authors have delared that no ompeting interests exist. Referenes [1] Mun J. Modeling risk: Applying Monte Carlo, risk simulation, strategi real options, stohasti foreasting, and portfolio optimization, nd Ed, Wiley & Sons; 010. ISBN: [] Paul M. Introdution to probability and statistial appliations. Addison-Wesley, Reading MA; ISBN: [3] Johnson D. The triangular distribution as a proxy for the beta distribution in risk analysis. The Statistiian. 1979;46: [4] David FN, Barton DE. Combinatorial hane. Lubreht & Cramer Ltd; 196. ISBN: [5] John R. Combinatorial hane. The Annals of Mathematial Statistis. 196;33(4): [6] Seal HL. The historial development of the use of generating funtions in probability theory. Mitt Verein Shweiz Versih Math. 1949;49:09-8. [7] Kotz S, Dorp JRV. Beyond beta. World Sientifi Publishing Co. Ltd, Singapore; 004. ISBN: [8] Jane M, Thomopoulos N. Min and max triangular extreme interval values and statistis. Journal of Business and Eonomis Researh. 010;8:

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11 British Journal of Mathematis & Computer Siene 4(4),.., 014 [5] Karlis D, ekalaki E. On some distributions stemming from the triangular distribution. Tehnial Report, Dept. of Statistis, Athens University of Eonomis. 000;111. [6] Sarireh M. Estimation of HD drilling time using deterministi and triangular distribution funtions. JETEAS. 013;4(3): [7] Johnson D. Triangular approximations for ontinuous random variables in risk analysis. JORS. 00;53: [8] Bak EW, Boles W, Fry G. Defining triangular probability distributions from historial ost data. Journal of Constrution Engineering and Management. 000;16(1):9-37. [9] Brizzi M. A skewed model ombining triangular and exponential features: The two-faed distribution and its statistial properties. Austrian Journal of Statistis. 006;35(4): [30] Gary M, Choudhary S, Kalla SL. On the sum of two triangular random variables. International Journal of Optimization: Theory, Methods and Appliations. 009;1(3): [31] Kkonendji CC, Senga Kiesse T, Zohi SS. Disrete triangular disributions and nonparametri estimation for probability mass funtion. Journal of Non-parametri Statistis. 007;19: Okagbue et al.; This is an Open Aess artile distributed under the terms of the Creative Commons Attribution Liense ( whih permits unrestrited use, distribution, and reprodution in any medium, provided the original work is properly ited. Peer-review history: The peer review history for this paper an be aessed here (Please opy paste the total link in your browser address bar)