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Laplace Distribution: Ordinary Differential Equations

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Laplace Distribution: Ordinary Differential Equations Proceedings of the World Congress on Engineering and Computer Science 2018 Vol I WCECS 2018, October 23-25, 2018, San Francisco, USA Laplace Distribution: Ordinary Differential Equations Hilary I. Okagbue, Member, IAENG, Pelumi E. Oguntunde, Abiodun A. Opanuga and Patience I. Adamu results for other distributions have been proposed, see [16- Abstract— Differential calculus was used in this paper to 29] for details. obtain the ordinary differential equations (ODE) of the probability functions of the Laplace distribution. The Laplace distribution is sometimes refers to as the double parameters and support that characterized Laplace exponential distribution because of its identical nature to the distribution inevitably determine the behavior, existence, uniqueness and solution of the ODEs. The method is exponential distribution. Earlier studied by [30] and recommended to be applied to other probability distributions extended to the multivariate Laplace distribution by [31] and probability functions not considered in this paper. and multivariate generalized Laplace distribution by [32]. Computer codes and programs can be used for the Some of the other modifications and extensions of the implementation. distribution include: three-parameter asymmetric Laplace distribution [33], skew Laplace distribution [34], beta Index Terms— Differential calculus, quantile function, hazard function, reversed hazard function, survival function, Laplace distribution [35], truncated skew-Laplace inverse survival function, probability density function, distribution [36], alpha–Skew–Laplace distribution [37]. Laplace. Others are: normal-Laplace distribution [38] and Semi-α- Laplace distributions [39]. Some of the areas of the distribution have been explored I. INTRODUCTION such as: goodness of fit tests and other statistical tests [40- ALCULUS in general and differential calculus in 43]; estimation of parameters [44-45]. C particular is often used in statistics in parameter and An important variant of the distribution is the log-Laplace modal estimations. The method of maximum likelihood is distribution whose logarithm is the Laplace distribution. an example. Notable among the applications are the works of [46] that Differential equations often arise from the understanding applied the distribution to the analysis of financial data and modeling of real life problems or some observed while [47] applied it to regression analysis. physical phenomena. Approximations of probability Differential calculus was used to obtain the results. functions are one of the major areas of application of calculus and ordinary differential equations in mathematical statistics. The approximations are helpful in the recovery of II. PROBABILITY DENSITY FUNCTION the probability functions of complex distributions [1-10]. The PDF of the Laplace distribution is given by; 1 x Apart from mode estimation, parameter estimation and fx( ) exp (1) approximation, probability density function (PDF) of 2bb distributions can be transformed as ODE whose solution x yields the respective PDF. Some of which are available: see exp if x [11-15]. 1 b The aim of this paper is to obtain homogenous ODE for fx() (2) 2b x the probability density function (PDF), Quantile function exp if x (QF), survival function (SF), inverse survival function b (ISF), hazard function (HF) and reversed hazard function Differentiate equation (2), to obtain the first order ODE; (RHF) of the Laplace distribution. This will also help to Case I, x : provide the answers as to whether there are discrepancies 1 x between the support of the distribution and the conditions fx( ) exp (3) necessary for the existence of the various ODE. Similar 2bb 1 x (4) fx( )2 exp Manuscript received February 9, 2018; revised March 14, 2018. This 2bb work was sponsored by Covenant University, Ota, Nigeria. H. I. Okagbue, P. E. Oguntunde, A. A. Opanuga and P. I. Adamu are The equation can only exists for b 0. with the Department of Mathematics, Covenant University, Ota, Nigeria. The first order ODE for the PDF of the Laplace distribution [email protected] for case I is given as; [email protected] abiodun.opanuga @covenantuniversity.edu.ng bf( x ) f ( x ) 0 (5) [email protected] ISBN: 978-988-14048-1-7 WCECS 2018 ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online) Proceedings of the World Congress on Engineering and Computer Science 2018 Vol I WCECS 2018, October 23-25, 2018, San Francisco, USA 1 ODE 1; f (0) exp , (6) 1 x 2bb fx( )3 exp Case II, x : 2bb (20) 1 x 11 x (7) fx( ) exp =2 exp 2bb b 2 b b 1 x (8) fx( ) 2 exp (21) 2bb The equation can only exists for (22) The first order ODE for the PDF of the Laplace distribution ODE 2; for case II is given as; 1 x bf( x ) f ( x ) 0 (9) fx( )3 exp 2bb 1 (23) f (0) exp , (10) 11 x 2bb = 2 exp Differentiate equations (4) and (8), to obtain the second b 2 b b order ODE; 1 f()() x f x (24) Case I, : b 1 x bf( x ) f ( x ) 0 (25) (11) fx( )3 exp 2bb 1 (26) f (0) 2 exp The equation can only exists for 2 bb Two ODEs can be acquired from the further breaking down Differentiate equations (11) and (19) to obtain the third of equation (11). These are listed as follows; ODE 1; order ODE; 1 x Case I, : fx( )3 exp = 2bb 1 x (12) (27) fx( )4 exp 11 x 2bb 2 exp b 2 b b The equation can only exists for 1 Three ODEs can be acquired from the further breaking f()() x f x (13) b2 down of equation (27). These are listed as follows; ODE 1; b2 f( x ) f ( x ) 0 (14) 1 x ODE 2; fx( )4 exp 2bb 1 x (28) fx( )3 exp 11 x 2bb =3 exp (15) b 2 b b 11 x = 2 exp 1 b 2 b b f()() x f x (29) b3 1 f()() x f x (16) b3 f( x ) f ( x ) 0 (30) b ODE 2; bf( x ) f ( x ) 0 (17) 1 x 1 fx(x ) 4 exp (18) 2 bb f (0)2 exp 2bb (31) 11 x =22 exp Case II, : b 2 b b 1 x 1 fx( ) exp (19) f()() x f x (32) 3 b2 2bb b 0. 2 The equation can only exists for b f ( x ) f ( x ) 0 (33) Two ODEs can be acquired from the further breaking down of equation (19). These are listed as follows; ISBN: 978-988-14048-1-7 WCECS 2018 ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online) Proceedings of the World Congress on Engineering and Computer Science 2018 Vol I WCECS 2018, October 23-25, 2018, San Francisco, USA ODE 3; Table 1: First to fifth order ODE for the PDF 1 x fx( )4 exp Case I, : Case II, : x 2bb (34) 11 x = 3 exp b 2 b b b2 f( x ) f ( x ) 0 b2 f( x ) f ( x ) 0 1 f()() x f x (35) b bf( x ) f ( x ) 0 (36) 1 4 v f (0)3 exp (37) b f ( x ) f ( x ) 0 bf( x )2 fbb ( x ) 0 b3 fv ( x ) f ( x ) 0 b3 fv ( x ) f ( x ) 0 Case II, : b2 fv ( x ) f ( x ) 0 1 x (38) v v fx( ) 4 exp bf( x ) f ( x ) 0 bf( x ) f ( x ) 0 2bb b5 fv ( x ) f ( x ) 0 b5 fv ( x ) f ( x ) 0 The equation can only exists for 4 v Three ODEs can be acquired from the breaking down of b f( x ) f ( x ) 0 equation (39). These are listed as follows; ODE 1; b3 fv ( x ) f ( x ) 0 b3 fv ( x ) f ( x ) 0 1 x 2 v b f( x ) f ( x ) 0 fx( ) 4 exp 2bb vv vv (39) bf( x ) f ( x ) 0 bf( x ) f ( x ) 0 11 x =3 exp b 2 b b The results are similar to the ones proposed in [16-29]. 1 f()() x f x (40) b3 III. QUANTILE FUNCTION The QF of the Laplace distribution is given by; b3 f( x ) f ( x ) 0 (41) +b ln(2p) derived if x ODE 2; Qp() (49) b ln(2(1-p)) derived if x 1 x fx( ) 4 exp Differentiate equation (49), to obtain the first order ODE; 2bb 2 (42) Case I, : b f( x ) f11 ( x ) 0 x Q( p ) b ln(2 p ) (50) =22 exp b 2 b b b Qp() (51) (43) p The equation can only exists for is bp0,0 1. (44) The first order ODE for the QF of the Laplace distribution ODE 3; for case I is given as; 1 x pQ3 ( p ) b 0 (52) b f( x ) f ( x ) 0 fx( ) 4 exp 2bb Qb(0.1) 1.60944 (53) (45) 11 x Case II, : = 3 exp b 2 b b Q( p )x b ln(2(1 p )) (54) 1 b f()() x f x (46) Qp() (55) b 1 p bf( x ) f ( x ) 0 (47) The equation can1 only exists for is f()() x f x 1 The first order2 ODE for the QF of the Laplace distribution f (0) exp (48) b 3 for case II is given as; b 0. 2bb 2 b f( x ) f ( x ) 0 The result from first order to fifth order ODE are listed in (1p ) Q ( p ) b 0 (56) Table (1); bfQb(0.1)( x ) f ( x ) 0.5878 0 (57) ISBN: 978-988-14048-1-7 WCECS 2018 ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online) Proceedings of the World Congress on Engineering and Computer Science 2018 Vol I WCECS 2018, October 23-25, 2018, San Francisco, USA Differentiate equations (51) and (55), to obtain the second Case II, : order ODEs; 2b Case I, : Qp() (74) x (1 p )3 b Qp() (58) The equation can only exists for p2 Two ODEs can be acquired from the further breaking down 1 Q()() p Q p (59) of equation (74).
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