# Introduction We begin with the Bernoulli's distribution. If ?? is a random variable associated with a random experiment on a set ?? with two possible outcomes, then ?? has a Bernoulli distribution ??(??) given by ??(??; ??) = ?? ?? (1 ? ??) 1??? ; ?? = 0,1 (1.1) where the parameter ?? is the probability of success. If we extend the range (domain) of the independent variable ?? to {0,1,2, . . . , ??} we have the Binomial distribution ??(??, ??) given by ??(??; ??, ??) = ? ?? ?? ??? ?? (1 ? ??) ????? ; ?? = 0,1,2, ? , ?? where the parameter ?? is the number of independent trials. Suppose we impose a further condition on the domain of ??, that is selection (sampling) is made without replacement, then the number of trials (is no longer independent) gives rise to Hypergeometric distribution ??(??, ??, ??) given by ?(??; ??, ??, ??) = ? ?? ?? ?? ?? ??? ?? ??? ? ? ?? ?? ? ?? ? ?? ? ?? (1.3) where ??, ??, ?? are fixed constants, ?? = ??????{0, ?? ? ?? + ??} and ?? = ??????{??, ??}. Now, suppose we decide to fix the number of successes we require in (1.2) and then observe the random number of trials needed to obtain this number of successes, then the random number ?? of trials required to obtain the first success has a geometric distribution given by ð??"ð??"(??; ??) = ???? ???1 ; ?? = 1, 2, 3, ? (1.4??) and if the random variable ?? is the number of failures before the occurrence of the first success, then we have ð??"ð??"(??; ??) = ???? ?? ; ?? = 0, 1, 2, 3, ? (1.4??) Observe that the geometric distributions in (1.4??) and (1.4??) are distributions of the number of independent Bernoulli trials required to obtain a single success. Hence, a further generalisation is to seek for the distribution of the random variable ?? on which the ????? success (?? > 1) occurs, such a distribution is called the negative binomial distribution ????(??, ??) and is given by ????(??; ??, ??) = ? ???1 ???1 ??? ?? ?? ????? ; ?? = ??, ?? + 1, ?? + 2, ?. (1.5??) and if the random variable ?? is the number of failures before the occurrence of the first ??th success, then we have ????(??; ??, ??) = ? ??+???1 ???1 ??? ?? ?? ?? ; ?? = 0,1,2, ?. (1.5??) One of the most important generalizations of (1.2) above is the discrete multivariate distribution function that belong to the (one dimensional) multinomial distribution ??(?? , ?? 1 , ? , ?? ?? ) is given by ??(?? 1 , ?? 2 , ? , ?? ?? ) = ? ?? ?? 1 ,?? 2 ,?,?? ?? ? ?? 1 ?? 1 ?? 2 ?? 2 ? ?? ?? ?? ?? ; (1.6) where ? ?? ?? ?? ??=1 = 1 and ??, ?? 1 , ?? 2 , ? , ?? ?? are the parameters. To mention but a few, the probability mass functions considered in (1.1) to (1.6) are often referred as the classical or standard discrete probability mass functions. However most of these standard ?????? is inadequacy in modeling different types of scenario. Consequent, in recent times, researchers have focused more on generalizingimproving with the aim of making the functions to be more adequate, that is seeking for a probability distribution functions that will accommodate and at the same time applicable in modeling different types of scenario which the former probability distribution functions could not handle. In order to improve on the discrete models (1.1) to (1.6) we consider some of the important contributors and their results in the sequel. Philippou and Muwafi (1982) introduced the ?? which gives rise to several studies of distribution of order ?? as contained in the reference (which reduce to the respective classical probability distribution for ?? = 1) some of these distributions are given by ??(??; ??, ??, ??) = ? ? ? ?? 1 + ?? 2 + ? + ?? ?? , ?? ?? 1 , ?? 2 , ? , ?? ?? , ?? ? ?? 1 ,?? 2 ,?,?? ?? ???1 ?? =1 ?? ?? ? ?? ?? ? ? ?? ?? ?? ??=1 ; ?? = 0,1, ? , ?? ?? ?? ?? (1.7) Where ?? 1 + 2?? ; ?? ? ?? (1.8) Where ?? 1 + 2?? 2 + ? + ???? ?? = ?? ? ??. ????(??; ??, ??) = ? ? ?? 1 +?? 2 +?+?? ?? ,???1 ?? 1 ,?? 2 ,?,?? ?? ,???1 ? ?? 1 ,?? 2 ,?,?? ?? ?? ?? ? ?? ?? ? ? ?? ?? ?? ??=1 ; ?? ? ???? (1.9) Where ?? 1 + 2?? 2 + ? + ???? ?? = ?? ? ????. Are the binomial, geometric, negative binomial distribution of order ?? respectively. The asymptotic properties of some of these distributions give rise to other important distributions as studied by Aki et al (1984), Feller (1956). In 1986, Panaretos and Evdokia improved on some of the above distributions, in particular (1.2) and (1.3) via sampling from an urn containing ?? white balls and ?? black balls. The following Hypergeometric distribution of order ?? was introduced. Assuming that ?? balls are drawn one at a time; Without replacement gives rise to; ? 1 (??; ??, ??, ??) = ? ? ? ?? 1 + ?? 2 + ? + ?? ?? , ?? ?? 1 , ?? 2 , ? , ?? ?? , ?? ? ?? 1 ,?? 2 ,?,?? ?? ???1 ?? =1 ?? ?? ?? ?? ?? ??=1 ? ?? ????? ?? ?? ?? ??=1 ? (?? + ??) (??) ; ?? = 0,1, ? , ?? ?? ?? ?? (1.10??) With replacement gives rise to ? 2 (??; ??, ??, ??) = ? ? ? ?? 1 + ?? 2 + ? + ?? ?? , ?? ?? 1 , ?? 2 , ? , ?? ?? , ?? ? ?? 1 ,?? 2 ,?,?? ?? ???1 ?? =1 ? ?? ?? + ?? ? ???? ?? ?? ?? ??=1 ? ?? ?? + ?? ? ? ?? ?? ?? ??=1 ; ?? = 0,1, ? , ?? ?? ?? ?? (1.10??) With replacement and addition of one ball of the same colour that was selected, before the next draw gives rise to ? 3 (??; ??, ??, ??) = ? ? ? ?? 1 + ?? 2 + ? + ?? ?? , ?? ?? 1 , ?? 2 , ? , ?? ?? , ?? ? ?? 1 ,?? 2 ,?,?? ?? ???1 ?? =1 ?? ?? ?? ?? ?? ??=1 ? ?? ????? ?? ?? ?? ??=1 ? (?? + ??) (??) ; ?? = 0,1, ? , ?? ?? ?? ?? (1.10??) With replacement and addition of ?? balls of the same color that was selected, before the next draw gives rise to ? 4 (??; ??, ??, ??) = ? ? ? ?? 1 + ?? 2 + ? + ?? ?? , ?? ?? 1 , ?? 2 , ? , ?? ?? , ?? ? ?? 1 ,?? 2 ,?,?? ?? ???1 ?? =1 ? ?? ?? ? ?? ?? ?? ?? ??=1 ? ? ?? ?? ? ????? ?? ?? ?? ??=1 ? ? ?? + ?? ?? ? ?? = 0,1, ? , ?? ?? ?? ?? (1.10??) Where ?? 1 + ?? 2 + ? + ?? ?? = ?? ? ???? ? ??, ?? (?? ) = ??(?? ? 1) ? (?? ? ?? + 1) ?? (?? ) = ??(?? + 1) ? (?? + ?? ? 1) In 1986, Panaretos and Evdokia introduced the Cluster Binomial Distribution as an improvement on the classical binomial distribution via sampling from an urn containing ?? labeled balls (?? = 1,2, ? ??) with ?? ?? denoting the probability that a ball bearing the number ?? will be drawn, such that ? ?? ?? ?? ??=1 = ??. Then, ?? = 1 ? ?? is the probability that a ball bearing a zero will be drawn. Let ?? be a random variable that count the sum of the numbers on the balls drawn. If the random variable ?? take the value ?? for the ?? balls drawn, ?? 1 bear the number 1, ?? 2 bear the number 2 and so on, ?? ?? bear the number ?? so that ? ???? ?? ?? ??=1 = ?? and each of the remaining ?? ? ? ?? ?? ?? ??=1 balls bear the zero. Then the is given by; (1.11) In an attempt to improve on the (one dimensional) multinomial distribution given in (1. the associated factorial ?? (??) ! by ?? (??) ! = ?. ?? ??=1 ? ?? ?? ?? ?? ?? ?? ?? =1 = ??? ?? ?? ?! Where ??? ?? ?? ?! = ??? ?? ?? ?1 1 ?! ??? ?? ?? ?1 2 ?! ??? ?? ?? ?1 3 ?! ? ??? ?? ?? ?1 ?? ?? ?! ?????? ?? (??) = ?? (??ð??"ð??"?? ???ð??"ð??"????) This discrete probability functions has been applied to certain parameter estimation problems in time series and contingency table analysis of arbitrary ??dimensional tables. However, if ?? = 1, we obtain multinomial distribution ??(??, ?? 1 , ? , ?? ?? ) given in (1.6). In 1756 (republished in 1967), Abraham De Moivre studied the probability distribution for a fair (balanced) ??-sided die tossed ?? number of times. Let ?? ?? (?? ) be a random variable that count the total score in n rolls of an ??-sided die, the following probability mass function was obtained ????? ?? (?? ) = ??? = 1 ?? ?? ? (?1) ?? ?? 1 ??=0 ? ?? ?? ?? ???1+??????? ???1 ?; 0 ? ?? ? (?? ? 1)?? (1.13) Where ?? 1 = ?????? ???, [ ?? ?? ]? and [ ?? ?? ] is the greatest integer function less than or equal to ?? ?? . The coefficient of 1 ?? ?? often denoted by ?? ?? (??, ??) have been studied in detail by Dafnis et al (2007), Freund (1956), who discussed their role in occupancy theory. In particular, ?? ?? (??, ??) can be interpreted as "the number of ways of putting ?? indistinguishable objects into ?? numbered boxes with each box containing at most ?? ? 1 objects. So that if ?? = 2 we have the standard binomial coefficient given by ?? 2 (??, ??) = ? ?? ?? ?; 0 ? ?? ? ??. A recurrence formula for computing ?? ?? (??, ??) is given by ?? ?? (??, ??) = ? ?? ?? (?? ? 1, ?? ? ??) ?? ?1 ?? =0 (1.14) One can easily see that for ?? = 2, this recursion reduces to the well-known classical binomial identity. The number ?? ?? (??, ??) has been used extensively in probability studies Ashok, et al ( 2011 # Notes we refer to Bondarenko (1993); Dafnis, et al (2007); Freund (1956); Gabai (1970); Ollerton and Shannon (1998, 2004 and the references therein. In 1995, Balasubramanian et al introduced the extended binomial distribution of order ?? with index ?? and parameter ?? as an improved version of the standard binomial distribution and the distribution studied by Abraham De Moivre in 1756 via considering ?? roll of an ?? sided die which is not necessarily fair (balanced) with face marked ?? (?? = 0,1,2, ? , ?? ? 1) and a turn-up side probability ?? ?? (? ?? ?? ?? ?1 ??=0 = 1) satisfying the condition ?? ?? ? ?? ?? = ?? ? ??. It was proved that if ?? ?? (?? ) is a random variable that count the total score in ?? rolls of an ??-sided die then the probability mass function (??????) is given by ????? ?? (?? ) = ??; ??? = ?(?1) ?? ?? 1 ??=0 ? ?? ?? ? ? ?? ? 1 + ?? ? ???? ?? ? 1 ? ?? ?? ?? (?? ?1)????? ; 0 ? ?? ? (?? ? 1)?? (1.15) Where ?? 1 = ?????? ???, [ # ?? ?? ]? and [ # ?? ?? ] is the greatest integer function less than or equal to ?? ?? . Observe that if the die is a fair one, then it implies that ?? = ? Ashok et al (2011) studied and derived a recursion formula for the probability distribution of the sum of rolling a fair dice (6-sided die) ?? times (which is equivalently to rolling ?? several dice once) which is given by; ?? ?? (??) = 1 6 ??? ?? ?1 (?? ? 1) + ?? ?? ?1 (?? ? 2) + ? + ?? ?? ?1 (?? ? 6)? ; ?? = 1,2, ? , ??; ?? ? [??, 6??] (1.16) In 2017, Okoli studied a (?? ? ?? + 1)-sided die with turn-up side probability denoted by ??(??, ??) = ?? ?? ?? ?? : ??, ?? = 1,2,3, ? , ??; ?? + ?? = ??; 0 ? ??, ?? ? 1. The following theorem was proved. ????? ?? (?? ,?? ) = ??; ??? = ?(?1) ?? ?? ??=0 ? ?? ?? ? ? ?? ? ???? ? 1 ?? ? 1 ? ?? ?? ?? ???? ??? ; ?? ? ?? ? ???? Where ?? = min ???, ? ????? ?? ??. It is important to note that a careful examination of these papers and related works in the literature that dealt with improvement of probability distribution, shows that the improvements, extensions, generalisations so achieved by these authors are mostly, at least in one of the following directions: (i) Addition of one or more parameters to the original probability function, (ii) Extension of the domain or space of the parameter(s), Motivated by the results of the research in this direction via the work of Abraham De Moivre (1756), Balasubramanian et al(1995), Ashok et al (2011) and Okoli (2017), we seek to derived a probability distribution of an arbitrary sides of a geometric figure indexed in a finite set of Arithmetic Sequence. This will take care of some of the computational inadequacies due to the works of Abraham De Moivre (1756), Balasubramanian et al (1995), Ashok et al (2011) and , in modeling the distribution of sides of geometric figure indexed in an arbitrary finite set of Arithmetic Sequence, which we shall illustrate in the sequel. # II. # Methodology We shall use the telling example that follows to compare the distribution studied by Balasubramanian et al (1995) and in modeling the distribution of a fair die. For illustrative purpose, Let ?? 2 (6,6) be the sum of scores obtained in the toss of a fair die twice, we wish to construct a probability table for the distribution of ?? 2 (6,6) . First, we consider the sample spaces given below from which we then give the probability table for the distribution of ?? 2 (6,6) . ?? ?? (??; ??) = 1 ?? ?? ?(?1) ?? ?? ??=0 ? ?? ?? ? ? ?? ? 1 + ?? ? ???? ?? ? 1 ? ; 0 ? ?? ? (?? ? 1)??, ?? = ?????? ???, ? ?? ?? ?? ?? ?? (??; ??) = 1 ?? ?? ?(?1) ?? ?? ??=0 ? ?? ?? ? ? ?? ? ???? ? 1 ?? ? 1 ? ; ?? ? ?? ? ????, ?? = ?????? ???, ? ?? ? ?? ?? ?? We begin with Balasubramanian ?????? in Aki et al, (1984) denoted by ?? ?? (??; ??). Observe that III). ????? 2 (6,6) = 9? = 4 36 Using ?? ?? (??; ??) = 1 ?? ?? ?(?1) ?? ?? ??=0 ? ?? ?? ? ? ?? ? 1 + ?? ? ???? ?? ? 1 ? ; 0 ? ?? ? (?? ? 1)?? ? ?? ?? (9; ??) = 1 6 2 ?(?1) ?? ?? ??=0 ? 2 ?? ? ? 10 ? 6?? 1 ? = 1 6 2 ?(?1) 0 ? 2 0 ? ? 10 ? 6 × 0 1 ? + (?1) 1 ? 2 1 ? ? 10 ? 6 1 ?? = 1 6 2 [10 ? 2 × 4] =2 36 ; ? ?? ?? (9; ??) ? ????? 2 (6,6) = 9? Now with the ?????? defined by Okoli (2017), ?? ?? (??; ??). Observe that (1). In ?? ?? (??; ??), ?? ? ?? ? ???? implies that 2 ? ?? ? 12 (since ?? = 2 and ?? = 6). This does agree with the range of ?? given in Table II for a fair die tossed twice. Which is not the case for ?? ?? (??; ??). (2). ?? ?? (??; ??), give accurate probability value(s) for each value(s) of ??. To see this, observe from the probability distribution table (Table III). ????? 2 (6,6) = 9? = 4 36 Using ?? ?? (??; ??) = 1 ?? ?? ?(?1) ?? ?? ??=0 ? ?? ?? ? ? ?? ? ???? ? 1 ?? ? 1 ? ; ?? ? ?? ? ???? ? ?? ?? (9; ??) = 1 6 2 ?(?1) ?? ?? ??=0 ? 2 ?? ? ? 8 ? 6?? 1 ? = 1 6 2 ?(?1) 0 ? 2 0 ? ? 8 ? 6 × 0 1 ? + (?1) 1 ? 2 1 ? ? 8 ? 6 1 ?? = 1 6 2 [8 ? 2 × 2] =4 36 ; ? ?? ?? (9; ??) = ????? 2 (6,6) = 9? # Notes Hence we conclude that the ?????? we defined ?? ?? (9; ??) is more practicable to work with than the one defined by Balasubramanian et al (1995) in modeling the distribution of sums of sides of a standard die. Since the standard die is indexed in the finite arithmetic sequence {1,2,3, ? , ??}. It is important to note that if we choose the finite arithmetic sequence {0,1,2,3, ? , ?? ? 1} for the indexing, then ?? ?? (??; ??) will no longer be adequate, rather ?? ?? (??; ??) will be suitable in modeling the distribution of sums of sides data. Thus, we have seen that the distribution studied by Balasubramanian et al (1995) and is rather restrictive and particularized, in the sense that ordinarily it cannot be use in modeling the distribution of sides of geometric figure indexed in an arbitrary finite set of Arithmetic Sequence. As a matter of fact, this constitute a major weakness which we shall address in the sequel. Let ??, ??, ?? ? ?, we now proceed to define a probability distribution that will be suitable in modeling the distribution of sides of geometric figure indexed in an arbitrary finite set of Arithmetic Sequence given by {??, ?? + ??, ?? + 2??, ? , ??} where ??, ?? and ?? denote the common difference, first and last term, this implies that our geometric figure is ? ????? ?? + 1?-sided. Thus, a typical sample space and sample space of sums of scores of such geometric figure tossed twice is given as Now to introduce a little more perturbation (unfairness) on this geometric figure we let ?? ? ? (where ?? is not necessarily equal to ??) and then defined the turn-up side probabilities as ??(??, ??) = ?? ?? ?? ?? : ??, ?? = ??, ?? + ??, ?? + 2??, ? , ??; ?? + ?? = ??; 0 ? ??, ?? ? 1. (2.1) Where ?? < ?? ? ?? Clearly the discrete probability distribution function associated with die models mentioned above is not adequate for modelling the distribution of sums of the turn-up It then follows that the first and second derivatives of the function ð??"ð??"(. ) are given by ð??"ð??" ? (??) = ???? ?????+?? ?? ???1 ? (?? + ??)?? ??+???1 + ???? ????? ?? ???1 (2.5) ð??"ð??" ?? (?? ) = ??(?? ? 1)?? ?????+?? ?? ???2 ? (?? + ?? ? 1)(?? + ??)?? ??+???2 + ??(?? ? 1)?? ????? ?? ???2 (2.6) Equation (2.5) is nonlinear function of ?? whose root can be determined by applying any of the iterative approximation formulas for finding the roots (zeros) of nonlinear equations. Since ?? ? (0,1) by definition, observe that ð??"ð??" ?? (??) < 0 ? ?? ? (0,1). Hence this implies that the function ð??"ð??"(. ) is strictly increasing for 0 ? ?? ? ?? ??,??,??,?? and strictly decreasing for ?? ??,??,??,?? ? ?? ? 1. Where ?? ??,??,??,?? is the zero of the function ð??"ð??" ? (. ), which in turn correspond to the turning (maximum) point of the function ð??"ð??"(. ). Consequently it follows that ð??"ð??"(. ) is monotone (sectionally) and unimodal with the mode occurring at the turning point ?? = ?? ??,??,??,?? [see Balasubramanian, et However, if in particular we take ?? = ??, then there exists ?? ?? ,??,?? for a balanced figure such that ?? = ?? ??,??,?? = ??. Then the normalization condition also reduces to ?? ?? (?? ?????+?? ? ?? ?????+?? ) = (?? ?? ? ?? ?? ) (2.7) Equation (2.1) to equation(2.7) implies the results of the authors mentioned in (??), (??), ?????? (??) above. We state the following theorems which unify the results of the authors: Balasubramanian, et al (1995); Okoli (2017); Okoli (2017) in the next section of this work as follows. # III. Main Results In this section, we now proceed to state some important theorem associated with turn-up side probability for the geometric figure described in equation (2.1) (table V) and their consequences. # Proof We expand (2.2) in ?? independent rolls of ( ????? ?? + 1)-sides as follows ?? ?? (??) = ? ? ? ? ?? ?? ? ???? ?? ? ?? ?1 ? ?? ?????+?? ?? ?????+?? ?? ?????+?? ? 1 ? ? ???? ?? ? ?? ? ? ? ? ?? = ?? ???? ?? ???? ? 1 ? ?? ?????+?? 1 ? ?? ?? ? ?? ; ?? = ???? ?? = ?? ???? ?? ???? (1 ? ?? ?????+?? ) ?? (1 ? ?? ?? ) ??? = ?? ???? ?? ???? ??(?1) ?? ?? ??=0 ? ?? ?? ? ?? (?????+??)?? ? ?(?1) ?? ? ??? ?? ? ?? ???? ? ??=0 = ?? ???? ?? ???? ??(?1) ?? ?? ??=0 ? ?? ?? ? ?? (?????+??)?? ? ?? ? ?? ? 1 + ?? ?? ? ?? ???? ? ??=0 ? = ? ?(?1) ?? ?? ??=0 ?? ???? ? ?? ?? ? ? ??=0 ? ?? ? 1 + ?? ?? ? ?? (?????+??)??+???? +???? = ? ?(?1) ?? ?? ??=0 ?? ???? ? ?? ?? ? ? ??=???? ? ?? ? 1 + ? ?? ?? ? ? ? ?? ? ?? + ?? ?? ? ?? ? ? ???? ?? ? ? ?? ?? ? ? ? ?? ? ?? + ?? ?? ? ?? ? ? ???? ?? ? ? ?? ?? = ? ?(?1) ?? ?? ??=0 ? ?? ?? ? ? ??=???? ? ?? ? 1 + ? ?? ?? ? ? ? ?? ? ?? + ?? ?? ? ?? ? ? ???? ?? ? ?? ? 1 ? ?? ?? ?? ???? ??? ?? ?? Where (?? ? ?? + ??)?? + ???? + ???? = ??. Thus, it follows from the last equation above that the probability mass function ?????? is given by. Observe that several other corollaries can be deduce from the theorems above which reduces to the results obtained in Ashok et al (2011); Balasubramanian, (1995); Okoli (2017); Okoli (2017) as special cases. Succinctly, it follows that; if ?? = 0, ?? = 1 and ?? = ?? = ?? ? 1, we obtain the results of Balasubramanian et al (1995), if ?? = 0, ?? = 1 and ?? ? ?? = ?? ? 1, we obtain the results of , if ?? = 1, ?? = 1 and ?? ? ?? = ??, we obtain the results of and if ?? = 1, ?? = 1 and ?? = ?? = 6, we obtain the results of Ashok et al (2011). Hence, the results of this research work unifies and improves the works of several researchers in this direction, haven shown that the existing results in the literature can be deduce easily from the results in this paper. # ?? ??? ?? ??? ?? ? ?????+?? ?? , ?????+?? ?? ? = ??; ??? = ?(?1) ?? ?? 3 ??=0 ? ?? ?? ? ? ?? ? 1 + ? ?? ?? ? ? ? ?? ? ?? + ?? ?? ? ?? ? ? ???? ?? ? ?? ? 1 ? ?? ?? ?? ???? ??? ; ???? ? ?? ? ?????? ??? ?? ? ?????+?? ?? , ?????+?? ?? ? = ??; ??? = ? ?? ?? ? ?? + ?? ? ?? ?(?1) ?? ?? 3 ??=0 ? ?? ?? ? ? ?? ? 1 + ? ?? ?? ? ? ? ?? ? ?? + ?? ?? ? ?? ? ? ???? ?? ? ?? ? 1 ? ; ???? ? ?? ? ???? Theorem 3.5 (??) ?? ?? ? (1) = ?? ??? ?? ? ?????+?? ?? , ?????+?? ?? ? ? = ???? + ???? ?? ? ?? ? (?? ? ?? + ??)?? ?? ?? (?????) ?? ?? ? ?? ?? ? (??) ?? ?? ?? (1) = ????(???? ? 1) + 2???? 2 ?? ?? ? ?? ? (?? ? ?? + ??)?? ?? ?? (?????) ?? ?? ? ?? ?? ? + ??(?? ? 1) ? ???? ?? ? (?? ? ?? + ??)?? ??+?? ?? (?????) ?? ?? ? ?? ?? ? 2 + ???? (?????) ? ?(?? ? ?? + ??)(?? ? ?? + ?? ? 1)?? ??+?? + ??(?? ? 1)?? ?? ?? (?????) ?? ?? ? ?? ?? + 2?? 2 ?? 2?? ?? (?????) ? 2??(?? ? ?? + ??)?? ??+2?? (?? ? ??) 2 ? Proof. Since ?? ?? (??) = ?? (?????)?? ?? ???? ?? ???? ? ?? ????? +?? ??? ????? +?? ?? ????? +?? ?? ?? ??? ?? ?? ?? ? ?? , it follows that the derivative ?? ?? ? (??) of ?? ?? (??) is (i) ?? ?? ? (??) = ?????? (?????)?? ?? ???? ?? ???? ?1 ? ?? ?????+?? ?![); Balasubramanian, et al (1995); De Moivre (1756); Feller (1968); Makri and Philippou (2005); Makri et al (2007a; 2007b) and related areas like reliability and inferential statistics, Ailing (1993); Bollinger and Burchard (1990); Gabai (1970). For more properties on ?? ?? (??, ??); generalized Pascal triangles or Pascal triangles of order ??, Probability Distribution of Sum of Sides of a Geometric Figure Indexed in Arithmetic Sequence](image-2.png "") 111![?? so that on substitution into equation (1.15) yield the result of Abraham De Moivrein in equation (1.13).](image-3.png "1 ?? ? 1 ?? ? 1 =") 1![?? ?? (?? ,?? ) be a random variable that count the total score in n rolls of an ??sided die with range ?? = 1,2, ? , ?? and turn-up side probabilities ??(??, ?? ? ??) (?? ? {1,2,3, ? , ??}) satisfying the condition ??(?? ?? ? ?? ?? ) = (?? ? ??) then the probability mass function (??????) is given by](image-4.png "Theorem 1. 1 Let") ![JournalsProbability Distribution of Sum of Sides of a Geometric Figure Indexed in Arithmetic Sequence Notes (iii) Extension of the domain or dimension of the independent variable of the original probability function.](image-5.png "") ![Journals Probability Distribution of Sum of Sides of a Geometric Figure Indexed in Arithmetic Sequence 1](image-6.png "") 223![With the normalization condition?? ?? (?? ?????+?? ? ?? ?????+?? ) = ?? ????? (?? ?? ? ?? ?? ) (2If for a fix ??, we define the auxiliary function ð??"ð??"(??) = 0 byð??"ð??"(??) = ?? ?? (?? ?????+?? ? ?? ?????+?? ) ? ?? ????? (?? ?? ? ?? ?? ) (2. 4)](image-7.png "( 2 . 2 ). 3 )") ![al (1995); Dharmadhikari and Joag-dev (1988); Hogg and Craig (1978); Okoli et al (2016); Okoli (2017); Okoli (2017)].](image-8.png "") 1![variable that count the total score in n rolls of a ( ????? ?? + ??)-sided geometric figure with turn-up side probabilities ??(??, ?? ? ??) satisfying the condition ?? ?? (?? ?????+?? ? ?? ?????+?? ) = ?? ????? (?? ?? ? ?? ?? ) , with range ?? = ??, ?? + ??, ?? + 2??, ? , ??. Then the probability generating function (??ð??"ð??"??) is given by ?? ?? (??) = ?? ??? ?? ?? ? ????? +?? ?? , ????? +?? ?? ? ? = ??? ????? ?? ?? ?? ?? ??? ????? +?? ??? ????? +?? ?? ????? +?? ? ?? ?? ??? ?? ?? ?? distributed (iid) random variables corresponding to the scores of ? ????? ?? + 1?-sided die and turn-up side probabilities (??, ?? ? ??) . Thus ?? ?? (??) = ?? ??? ??? ????? ?? ?? ?? ?? (?? ?????+?? ? ?? ?????+?? ?? ?????+?? ) ?? ?? ? ?? ?? ?? ?? variable that count the total score in n rolls of a ( ????? ?? + 1)-sided geometric figure with turn-up side probabilities ??(??, ?? ? ??) satisfying the condition ?? ?? (?? ?????+?? ? ?? ?????+?? ) = ?? ????? (?? ?? ? ?? ?? ) , with range ?? = ??, ?? + ??, ?? + 2??, ? , ??. Then the probability mass function (??????) is given by ?? ?? ?? ???? ??? ; ???? ? ?? ? ???? Where ?? 3 = min ???, ? ???? ????? ?????+?? ?? , ?? ? ??.](image-9.png "= 1 ?") 11![?? ?? ?? ???? ??? ; ???? ? ?? ? ???? If we are dealing with a fair (balanced) die (i.e. ?? = ??, ?? = ? ?? ?????+?? ? ) then the corollary that follows is a consequence of theorem 2.2 above. variable that count the total score in n rolls of a ( ????? ?? + 1)-sided geometric figure with turn-up side probabilities ??(??, ?? ? ??) satisfying the condition ?? ?? (?? ?????+?? ? ?? ?????+?? ) = (?? ?? ? ?? ?? ) , with range ?? = ??, ?? + ??, ?? + 2??, ? , ??. Then the probability mass function (??????) is given by](image-10.png "1 ? 1 ??") ![Probability Distribution of Sum of Sides of a Geometric Figure Indexed in Arithmetic Sequence Notes variable that count the total score in n rolls of a ( ????? ?? + 1)-sided geometric figure with turn-up side probabilities ??(??, ?? ? ??) satisfying the condition ?? ?? (?? ?????+?? ? ?? ?????+?? ) = (?? ?? ? ?? ?? ) , with range ?? = ??, ?? + ??, ?? + 2??, ? , ??. Then the probability mass function (??????) is given by](image-11.png "") ?? 1 ,?? 2 ,?,?? ????? ?? ? ?? ?? ? ? ?? ??=1?? ?? Probability Distribution of Sum of Sides of a Geometric Figure Indexed in Arithmetic Sequence???? ????? (??) ? = ?.? ?? ?? ????=1?? ?? =1the associated monomial ?? ?? (?? ) by???? ???? ?? (?? ) = ?. ??=1?? ?? ?? ? ?? ?? ?? ?? ?? =1NotesYear 20196), Okoli et al introduce the following parameter; let be a multi-index (or multi-integer), ?? {?? 1 , ?? 2 , ? , ?? ?? } ?? 2 , ? , ?? ?? ?? 1 , ?? 2 ??? 1 ?? ?? ? a finite multi-set induced by ?? = (?? 1 , ?? 2 , ? , ?? ?? ). However, he observed ??(??, ??) = that for arbitrary but fixed ?? ? ?, the multinomial distributions in (1.6) do not give adequate description to many important practical problems defined on the more general set given by ?????, ?? (??) )? = ??? ?? ?? ?? ?? ?? ? ?? ?? ? [?? ?? ], ?? ?? ? ?, ?? ? [??] ? (1.12) where ?? (??) = ??? ?? ?? ? ?? ?? ? [?? ?? ], ?? ?? ? ?, ?? ? [??]?, [?? ?? ] = {1,2,3, ? , ?? ?? }. Theorem 1.0 Let ?? (??) (??; ?? ??=1 ?? ? ??? ?? ???1 ? ? ? ??? ?? ???1 ?? ?? ? ?? ?? ?1 ?? ?? =1 ??? ?? ???1 ?? ?? ? ? ?? ?? ?? ?? =1 ? ? ?? ?? ?? ?? ?? ?? ?? ?? Where??? ?? ???1 ? ? ? ??? ?? ???1 ?? ?? ? ?? ?? ?? ?? =1 ?????? ????(??; ??, ??, ?? 1 , ? , ?? ?? ) = ? ? ?? ?? 1 , ?? 2 , ? , ?? ?? , ?? ? ? ?? ?? ?? ??=1 ? ?? 1 ,?? 2 ,?,?? ?? ?? ?? ?? ?? ?? ?? ??=1 ? ?? ???? ?? ?? ?? ??=1 ??(?? , ?? 1 ,?,?? ?? ) ??=(?? 1 , ?? 2 , ? , ?? ?? ) = a finite set and = 0 ? ?? ? [??], 1 Global Journal Frontier Research Volume XIX Issue ersion I V II 22 ( F ) For such case, more adequate and elaborate discrete distribution models are of Science needed which they proved in the theorem that follows© 2019 Global Journals I: (Sample space of twice tossed die)Table II: (Sample space of sum of scores) Table III: (Probability distribution table) Now let ?? ?? (??) and ?? ?? (??) denotes the probability mass functions due to Balasubramanian et al (1995) and Okoli (2017); that is Notes2 3 4 5 6 73 4 5 6 7 84 5 6 7 8 95 6 7 8 9 106 7 8 9 10 117 8 9 10 11 12??2345678910 11 12(6,6) = ??) 1 ??(?? 2 362 363 364 365 366 365 364 363 362 361 36(1) In ?? ?? (??; ??), 0 ? ?? ? (?? ? 1)?? implies that 0 ? ?? ? 10 (since ?? = 2 and ?? = 6). Year 20191 26ersion I VIIIssueVolume XIX( F )Frontier Researchof ScienceGlobal Journal© 2019 Global Journals IVarithmetic sequence)(Sample space of sums of scores for the geometric figure indexed in arithmeticsequence) Probability Distribution of Sum of Sides of a Geometric Figure Indexed in Arithmetic SequenceNow, observe that the generating function ??(??) for the ? given by????? ?? + 1?-sided figure is??(??) =NotesYear 20191 27ersion I VIIIssue??, ?? (?? + ??), ????, (?? + ??)??, (?? + 2??)? ???, ?? (?? + 2??), ??Volume XIXTable V:? ??, ?? 2?? + ?? 2??? ??, (?? + ??) 2?? + ?? 2?? + 2??? ??, (?? + 2??) 2?? + 2?? ? 2?? + 3?? ?? ? ?? + ?? ?? + ?? + ??? ??, ??F ) ( Frontier Research2?? + 2?? ? ?? + ??2?? + 3?? ?2?? + 4?? ?? ?2?? + ?? + ?? ?of ScienceGlobal Journal© 2019 Global Journals(?? + ??), (?? + ??) (?? + ??), (?? + 2??) ? (?? + ??), ?? (?? + 2??), ?? (?? + 2??), (?? + ??) 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