Probability Distribution Relationships - IJENS

1 International Journal of Basic & Applied Sciences IJBAS- IJENS Vol:10 No:01 48. Probability Distribution Relationships Yousry H. Abdelkader and Zainab A. Al-Marzouq Dept. of Math., Faculty of Science , Alexandria University, Egypt Dept. of Math., Girls College, King Faisal University, KSA. E-mail address: E-mail address: Abstract In this paper, we are interesting to show the most The equation (1) may be linear or non-linear equation. In the famous distributions and their relations to the other distributions case of linearity, it could be taken the form in collected diagrams. Four diagrams are sketched as networks. n The first one is concerned to the continuous distributions and Y ai X i their relations. The second one presents the discrete distributions. i 1. The third diagram is depicted the famous limiting distributions.

2 Many distributions, for this linear transformation, give the Finally, the Balakrishnan skew-normal density and its relationship with the other distributions are shown in the fourth same distributions for different values for ai such as: normal, diagram. gamma, chi-square and Cauchy for continuous distributions and Poisson, binomial, negative binomial for d iscrete Index Term-- Probability Distributions, distributions as indicated in the Fig. by double rectangles. On Transformations, Limiting Distributions. the other hand, when ai 1 , the equation (3) gives another I. INT RODUCT ION Distribution , for examp le, the sum of the exponential 's gives the Erlang Distribution and the sum of geomet ric 's In spite of the variety of the Probability d istributions, many of gives negative- binomial d istribution as well as the sum of them are related to each other by different relat ions hips.

3 Bernoulli 's gives the binomial Distribution . Moreover, the Deriving the Probability d istribution fro m other Probability difference between two 's give another Distribution , for distributions are useful in different situations, for example, example, the d ifference between the exponential 's gives parameter estimations, simu lation, and finding the Probability Laplace Distribution and the difference between Poisson 's of a certain Distribution depends on a table of another gives Skellam Distribution , see Fig. 1 and 2. Distribution . The Relationships among the Probability In the case of non-linearity of equation (1) , the derived distributions could be one of the two classifications: the transformations and limiting distributions. In the Distribution may give the same Distribution , for example, the transformations, there are three most popular techniques for product of log-normal and the Beta distributions give the same finding a Probability Distribution fro m another one.

4 These Distribution with different parameters; see, for examp le, Cro w three techniques are: and Shimizu (1988), Kotlarski (1962), and Krysicki (1999). On the other hand, equation (1) may be give d ifferent 1- The cumulative Distribution function technique, Distribution as indicated in the 2- The transformation technique, and The other classificat ion is the asymptotic or 3- The moment generating function technique. approximating d istributions. The asymptotic theory or limiting The main idea of these techniques works as follows: For given Distribution provides in some cases exact but in most cases functions g i (X 1 , X 2 , .., X n ) , for i 1, 2, , k where approximate d istributions. These approximations of one Distribution by another one exist. Fo r example, for large n the joint Distribution of random variables ( 's ).

5 And small p the binomial Distribution can be appro ximated X 1 , X 2 , .., X n is given, we define the functions by the Poisson Distribution . Other approximations can be given by the central limit theorem. For examp le, for large n Y i g i (X 1 , X 2 , .., X n ), i 1, 2, .., k and constant p , the (1) central limit theorem gives a normal The joint Distribution of Y 1 ,Y 2 , ..,Y n can be determined by approximation of the binomial d istribution. In the first case, one of the suitable method sated above. In particular, for the binomial Distribution is discrete and the appro ximating Poisson Distribution is also discrete. While, in the second case, k 1 , we seek the Distribution of the binomial Distribution is discrete and the appro ximating Y g (X ) (2). normal d istribution is continuous.

6 In most cases, the normal or standard normal p lays a very predo minant ro le in other For some function g (X ) and a given X . distributions. 1001-91310-3434 IJBAS- IJENS February 2010 IJENS . IJEN S. International Journal of Basic & Applied Sciences IJBAS- IJENS Vol:10 No:01 49. The most important use of the Relationships between the Throughout the paper, the words "diagram" and " fig." shall be Probability distributions is the simulation technique. Many of used synonymously. the methods in computational statistics require the ability to generate random variables fro m known Probability II. THE M AIN FEAT URES OF T HE FIGURES. distributions. The most popular method is the inverse Many distributions have their genesis in a prime transformation technique wh ich deals with the cu mu lative Distribution , for examp le, Bernoulli and uniform distributions Distribution function, F (x ) , of the Distribution to be form the bases to all distributions in discrete and continuous simulated.

7 By setting case, respectively. The main features of Fig. 1 exp lain the continuous F (x ) U Distribution Relationships using the transformat ion techniques. Where F (x ) and U are defined over the interval (0,1) These transformations may be linear or non-linear. The and U is a follows the uniform Distribution . Then, x is uniform Distribution forms the base to all other distributions. uniquely determine by the relation The Appendix contains the well known d istributions which are used in this paper and are obtained fro m the following web x F 1 (U ) (4) site: m. It is written in the Appendix in concise and compact way. We do not present Unfortunately, the inverse transformation technique can not be proofs in the present collection of results. For surveys of this used for many distributions because a simple closed form materials and additional results we refer to Johnson et al.

8 Solution of (4) is not possible or it is so complicated as to be (1994) and (1995). impractical. When this is the case, another Distribution with a The main features of Fig. 2 can be expressed as follows. simple closed form can be used and derived from another or If X 1 , X 2 , .. is a sequence of independent Bernoulli 's, other distributions. For example, to generate an Erlang deviate we only need the sum m exponential deviates each with the number of successes in the first n trials has a binomial Distribution and the number of failures before the first success expected value 1 m . Therefore, the Erlang variate x is has a geometric Distribution . The number o f failure before the expressed as kth success (the sum of k independent geometric 's) has a m 1m x yi lnU i , Pascal or negative bino mial Distribution .

9 The samp ling without replacement, the number of successes in the first n i 1 i 1 trials has a hyper-geometric Distribution . Moreover, the Where y i is an exponential deviate with parameter , exponential Distribution is limit of geo metric d istribution, and the Erlang Distribution is limit of negative binomial generated by the inverse transform technique and U i is a Distribution . The other Relationships of discrete Distribution as random nu mber fro m the uniform Distribution . Therefore, a well as the analogue continuous Distribution can be seen complicated situation as in simulat ion models can be replaced clearly in Fig. 2. by a comparatively simp le closed form Distribution or Fig. 3 is depicted the asymptotic d istributions together asymptotic model if the basic conditions of the actual situation with the conditions of limiting.

10 Limit ing distributions, in Fig. are compatible with the assumptions of the model. 3 and 4, are indicated with a dashed arrow. The standard Leemis (1986), Taha (2003) and Rider (2004) have tried to normal and the binomial d istribution play a very predominant figure the Relationships amog the Probability d istributions in role in other distributions. limited attempts. The first and second authors have presented Sharafi and Behboodian (2008) have introduced the a diagram to show the relat ionships among Probability Balakrishnan skew-normal density ( SNB n ( ) ) and studied distributions. The diagrams have twenty eight and nineteen distributions including: continuous, discrete and limiting its properties. They defined the SNB n ( ) with integer distributions, respectively.