PROBABILITY DISTRIBUTION RELATIONSHIPS. Y.H. Abdelkader, Z.A. Al-Marzouq 1. INTRODUCTION

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1 STATISTICA, ao LXX,., 00 PROBABILITY DISTRIBUTION RELATIONSHIPS. INTRODUCTION I spite of the variety of the probability distributios, may of them are related to each other by differet kids of relatioship. Derivig the probability distributio from other probability distributios are useful i differet situatios, for eample, parameter estimatios, simulatio, ad fidig the probability of a certai distributio depeds o a table of aother distributio. The relatioships amog the probability distributios could be oe of the two classificatios: the trasformatios ad limitig distributios. I the trasformatios, there are three most popular techiques for fidig a probability distributio from aother oe. These three techiques are: - The cumulative distributio fuctio techique - The trasformatio techique 3 - The momet geeratig fuctio techique. The mai idea of these techiques works as follows: For give fuctios gi( X, X,..., X, for i,,..., k where the joit distributio of radom variables (r.v. s X, X,..., X is give, we defie the fuctios Y g ( X, X,..., X, i,,..., k ( i i The joit distributio of Y, Y,..., Y ca be determied by oe of the suitable method sated above. I particular, for k, we seek the distributio of Y g( X ( For some fuctio g( X ad a give r.v. X. The equatio ( may be liear or o-liear equatio. I the case of liearity, it could be take the form Y a X (3 i i i

2 4 May distributios, for this liear trasformatio, give the same distributios for differet values for a i such as: ormal, gamma, chi-square ad Cauchy for cotiuous distributios ad Poisso, biomial, egative biomial for discrete distributios as idicated i the Figures by double rectagles. O the other had, whe ai, the equatio (3 gives aother distributio, for eample, the sum of the epoetial r.v. s gives the Erlag distributio ad the sum of geometric r.v. s gives egative- biomial distributio as well as the sum of Beroulli r.v. s gives the biomial distributio. Moreover, the differece betwee two r.v. s give aother distributio, for eample, the differece betwee the epoetial r.v. s gives Laplace distributio ad the differece betwee Poisso r.v. s gives Skellam distributio, see Figures ad. I the case of o-liearity of equatio(, the derived distributio may give the same distributio, for eample, the product of log-ormal ad the Beta distributios give the same distributio with differet parameters; see, for eample, (Crow ad Shimizu, 988, (Kotlarski, 96, ad (Krysicki, 999. O the other had, equatio ( may be give differet distributio as idicated i the Figures. The other classificatio is the asymptotic or approimatig distributios. The asymptotic theory or limitig distributio provides i some cases eact but i most cases approimate distributios. These approimatios of oe distributio by aother oe eist. For eample, for large ad small p the biomial distributio ca be approimated by the Poisso distributio. Other approimatios ca be give by the cetral limit theorem. For eample, for large ad costat p, the cetral limit theorem gives a ormal approimatio of the biomial distributio. I the first case, the biomial distributio is discrete ad the approimatig Poisso distributio is also discrete. While, i the secod case, the biomial distributio is discrete ad the approimatig ormal distributio is cotiuous. I most cases, the ormal or stadard ormal plays a very predomiat role i other distributios. The most importat use of the relatioships betwee the probability distributios is the simulatio techique. May of the methods i computatioal statistics require the ability to geerate radom variables from kow probability distributios. The most popular method is the iverse trasformatio techique which deals with the cumulative distributio fuctio, F (, of the distributio to be simulated. By settig F ( U Where F ( ad U are defied over the iterval (0, ad U is a r.v. follows the uiform distributio. The, is uiquely determie by the relatio F ( U (4 Ufortuately, the iverse trasformatio techique ca ot be used for may distributios because a simple closed form solutio of (4 is ot possible or it is

3 Probability distributio relatioships 43 so complicated as to be impractical. Whe this is the case, aother distributio with a simple closed form ca be used ad derived from aother or other distributios. For eample, to geerate a Erlag deviate we oly eed the sum m epoetial deviates each with epected value m. Therefore, the Erlag variate is epressed as m m y lu i i i i Where y i is a epoetial deviate with parameter, geerated by the iverse trasform techique ad U i is a radom umber from the uiform distributio. Therefore, a complicated situatio as i simulatio models ca be replaced by a comparatively simple closed form distributio or asymptotic model if the basic coditios of the actual situatio are compatible with the assumptios of the model. The relatioships amog the probability distributios have bee represeted by (Leemis, 986, (Taha, 003 ad (Rider, 004 i limited attempts. The first ad secod authors have preseted a diagram to show the relatioships amog probability distributios. The diagrams have twety eight ad ietee distributios icludig: cotiuous, discrete ad limitig distributios, respectively. The Rider s diagram divided ito four categories: discrete, cotiuous, semi-bouded, ad ubouded distributios. The diagram icludes oly twety distributios. This paper presets four diagrams. The first oe shows the relatioships amog the cotiuous distributios. The secod diagram presets the discrete distributios as well as the aalogue cotiuous distributios. The third diagram is cocered to the limitig distributios i both cases: cotiuous ad discrete. The Balakrisha skew-ormal desity ad its relatioships with other distributios are show i the fourth diagram. It should be metioed that the first diagram ad the fourth oe are coected. Because the fourth diagram depeds o some cotiuous distributios such as: stadard ormal, chi-square, the stadard Cauchy, ad the studet s t-distributio. Throughout the paper, the words diagram ad figures shall be used syoymously.. THE MAIN FEATURES OF THE FIGURES May distributios have their geesis i a prime distributio, for eample, Beroulli ad uiform distributios form the bases to all distributios i discrete ad cotiuous case, respectively. The mai features of Figure eplai the cotiuous distributio relatioships usig the trasformatio techiques. These trasformatios may be liear or o-liear. The uiform distributio forms the base to all other distributios.

4 44 The Appedi cotais the well kow distributios which are used i this paper ad are obtaied from the followig web site: wolfram.com. It is writte i the Appedi i cocise ad compact way. We do ot preset proofs i the preset collectio of results. For surveys of this materials ad additioal results we refer to (Johso et al., 994 ad 995. Figure Cotiuous distributio relatioships. The mai features of Figure ca be epressed as follows. If X, X,... is a sequece of idepedet Beroulli r.v. s, the umber of successes i the first trials has a biomial distributio ad the umber of failures before the first success has a geometric distributio. The umber of failure before the kth success (the sum of k idepedet geometric r.v. s has a Pascal or egative biomial distributio. The samplig without replacemet, the umber of successes i the first trials has a hyper-geometric distributio. Moreover, the epoetial distributio is limit of geometric distributio, ad the Erlag distributio is limit of egative biomial distributio. The other relatioships of discrete distributio as well as the aalogue cotiuous distributio ca be see clearly i Figure.

5 Probability distributio relatioships 45 Figure Beroulli s trials ad its related distributios. Figure 3 is depicted the asymptotic distributios together with the coditios of limitig. Limitig distributios, i Figure 3 ad 4, are idicated with a dashed arrow. The stadard ormal ad the biomial distributio play a very predomiat role i other distributios. Figure 3 Limitig distributios.

6 46 (Sharafi ad Behboodia, 008 have itroduced the Balakrisha skewormal desity ( SNB ( ad studied its properties. They defied the SNB ( with iteger by f ( ; c ( ( (,,, (5 where the coefficiet c ( is give by c(, ( ( d E( ( U where U N(0,. A special case arises whe ad c( which gives the skew-ormal desity ( SN(, see (Azzalii, 985. The probability distributio of the Balakrisha skew-ormal desity ad its relatioships of the other distributios are also discussed. The most importat properties of SNB ( are: ( i SNB ( is strogly uimodal, ( ii c(, c ( si, 4 3 c3( si, 8 4 Where is deoted the correlatio coefficiet. For 4, there is o closed form for c(. But some approimate values ca be foud i (Steck, 96. The bivariate case of SNB ( ad the locatio ad scale parameters are also preseted. Figure 4 eplais the mai features of their results. The dotted arrows idicate asymptotic distributio.

7 Probability distributio relatioships 47 Figure 4 The balakrisha skew-ormal desity SNB (. 3. CONCLUDING REMARKS It is a reasoable assertio that all probability distributios are someway related to oe aother. I this paper, four diagrams summarize the most popular relatioships amog the probability distributios. The relatioships amog the probability distributios are oe of the two classificatios: trasformatio ad limitig. Each diagram eplais itself. The advatages of usig these diagrams are: the studet at the seior udergraduate level or begiig graduate level i statistics or egieerig ca use the diagrams to supplemet course material. Besides, the researchers ca use the diagrams for fastig search for the relatioships amog the distributios. These diagrams are just start poits. Similar diagrams ca be costructed to summarize may statistical theorems such as: the characterizatios of distributios based o; order statistics, Records ad Momets, see (Gather et al., 998. Departmet of Math. Faculty of Sciece, Aleadria Uiversity, Egypt Departmet of Math. College, Kig Faisal Uiversity, KSA. YOUSRY ABDELKADER ZAINAB AL-MARZOUQ

8 48 APPENDIX TABLE Cotiues distributio Beta Distributio Domai [0,] Parameters 0, 0 PDF ( B(, Beta Prime Distributio [0, 0, 0 ( B(, Cauchy Distributio (,, b 0 b ( b Chi Distributio [0, 0 ( e Chi-Squared Distributio [0, 0 e ( Degeerate Distributio { 0} 0, ( 0 0, 0 0 Erlag Distributio [0,,, 3,, 0 e (! Epoetial Distributio [0, 0 f ( e F- Distributio [0, 0, 0, ( B Gamma Distributio Gibrat s Distributio [0, (0, 0, 0 Noe e ( e (l / Gumbel Distributio Half-Normal Distributio (, [0, 0, h 0 e e h e h / Iverse Chi-Square Distributio (0, 0 f ( e Iverse-Gamma Distributio (0, 0, 0 f ( e Kumaraswamy Distributio Laplace Distributio Levy Distributio [0,] (, [0, a 0, b 0 b 0, c 0 a a b ab ( e b c c e 3/ b

9 Probability distributio relatioships 49 Logistic Distributio Logormal Distributio (, [0, b 0, 0, b e b b e [l ] ep Mawell Distributio [0, a 0 a 3 e a Nocetral Chi-Squared Distributio [0, 0, 0 k e ( k 0! ( k k k Nocetral F-Distributio [0,,,, 0 k l k k l k0l0 kl e Bk, l ( kl ( Nocetral Studet s t-distributio (, 0, 0! / e ( 3 F ; ; F ; ; ( ( ( Normal Distributio (, 0, e ( Pareto Distributio [ 0, 0 0, k0 k k0 k Pearso Type III Distributio [0,, 0, e ( Rayleigh Distributio [0, 0 e Rice Distributio [0, 0, 0 o e l Stadard Normal Distributio Studet s t-distributio (, (, Noe 0 f ( e ( / Triagular Distributio Uiform Distributio Wald Distributio [ ab, ] [ ab, ] (0, a c b ab, 0, 0 ( a a c ( b a( c a ( b c b ( b a( b c b a / ( ep 3 Weibull Distributio [0,, 0 e

10 50 TABLE Discrete distributio Beroulli Distributio Beta Biomial Distributio Biomial Distributio Discrete Uiform Distributio Geometric Distributio Hypergeometric Distributio Domai {0,} {0,,..., } {0,,..., } {0,,..., } {0,,,...} {0,,..., } Parameters 0 p, q p,, 0,,,... 0 p, q p,,,...,,... 0 p, q p, N 0,,,..., K 0,,..., N, 0,,..., N, k p, q p N P.M.F. q 0 P ( p B (, P ( B(, P ( p q P ( P ( pq Np Nq P ( N Log-Series Distributio Pascal Distributio {,, 3,...} {0,,,...} 0 p 0 p, q p, k,,... P ( l( k k P ( p q k Poisso Distributio Rademacher Distributio Skellam Distributio {0,,,...} {,} {...,,0,,...} 0 Noe 0, 0 e P (! P ( / ( P ( e I REFERENCES A. AZZALINI (985, A class of distributios with icludes the ormal oes, Scadiavia joural of statistics,, pp E. L. CROW, K. SHIMIZU (988, Logormal distributios: Theory ad Applicatios, Marcel Dekker, New York. U. GATHER, U., KAMPUS,adN. SCHWEITZER (998, Characterizatios of distributios via idetically distributed fuctios of order statistics, I: Balakrisha, N., ad Rao, C. ed., Hadbook of Statistics, 6, Elsevier Sciece, pp N. JOHNSON, S. KOTZ ad N. BALAKRISHNAN (994, Cotiuous uivariate distributios, Vol., d ed. Wiley, New York. N. JOHNSON, S. KOTZ ad N. BALAKRISHNAN (995, Cotiuous uivariate distributios, Vol., d ed., Wiley, New York. I. KOTLARSKI (96, O group of idepedet radom variables whose product follows the beta distributio, Collog. Math. IX Fasc.,, pp W. KRYSICKI (999, O some ew properties of the beta distributio, Statistics & Probability Letters, 4, pp

11 Probability distributio relatioships 5 L. LEEMIS (986, Relatioships amog commo uivariate distributios, The America Statisticia, 40, pp M. SHARAFI, J. BEHBOODIAN (008, The Balakrisha skew-ormal desity, Statistical Papers, 49, pp J.W. RIDER (004, Probability distributio relatioships, H. A. TAHA (003, Operatios Research: A itroductio, 3rd ed, Macmilla Publishig Co. Probability distributio relatioships SUMMARY I this paper, we are iterestig to show the most famous distributios ad their relatios to the other distributios i collected diagrams. Four diagrams are sketched as etworks. The first oe is cocered to the cotiuous distributios ad their relatios. The secod oe presets the discrete distributios. The third diagram is depicted the famous limitig distributios. Fially, the Balakrisha skew-ormal desity ad its relatioship with the other distributios are show i the fourth diagram.