Beta type 3 distribution and its multivariate generalization *.Abstract The beta type 3 distribution has been obtained from the beta type 1 distribution by means of variable transformation. The beta type 3 distribution which is defined on bounded interval Noun 1. bounded interval - an interval that includes its endpoints closed interval interval - a set containing all points (or all real numbers) between two given endpoints may serve as an alternative to the beta type 1 distribution for many practical applications. Several properties of the beta type 3 distribution and its relationship to the beta type 1 and type 2 distributions have been studied. A multivariate The use of multiple variables in a forecasting model. generalization gen·er·al·i·za·tion n. 1. The act or an instance of generalizing. 2. A principle, a statement, or an idea having general application. of the beta type 3 distribution has also been obtained and its properties have been derived. Keywords and Phrases: Beta distribution Not to be confused with Beta function. In probability theory and statistics, the beta distribution is a family of continuous probability distributions defined on the interval [0, 1] parameterized by two non-negative shape parameters, typically denoted by α and β. ; Gauss hypergeometric function In mathematics, a hypergeometric function can be:
1. Introduction The beta (type 1) distribution with parameters ([alpha], [beta]) is defined by the probability density function Probability density function The function that describes the change of certain realizations for a continuous random variable. (p.d.f.) [{B([alpha], [beta])}.sup.-1][u.sup.[alpha]-1][(1 - u).sup.[beta]-1], 0 < u < 1, (1.1) where [alpha] > 0, [beta], > 0, and B([alpha], [beta]) is the beta function This article is about the Euler beta function. There are separate articles on the Dirichlet beta function and on the beta-function (written with a hyphen) of physics. In mathematics, the beta function defined by B([alpha], [beta]) = [[GAMMA The way brightness is distributed across the intensity spectrum by a monitor, printer or scanner. Depending on the device, the gamma may have a significant effect on the way colors are perceived. ]([alpha])[GAMMA]([beta])]/[[GAMMA]([alpha] + [beta])]. This distribution has two parameters and yet a rich variety of shapes. Because the beta distribution is bounded on both sides, it is often used for representing processes with natural lower and upper limits. It is most useful for modeling proportion: by adjusting its parameters, one can achieve a desired density function having a domain on (0, 1). The beta distribution is well known in Bayesian Adj. 1. Bayesian - of or relating to statistical methods based on Bayes' theorem methodology as a prior distribution on the success probability of a binomial distribution binomial distribution n. The frequency distribution of the probability of a specified number of successes in an arbitrary number of repeated independent Bernoulli trials. Also called Bernoulli distribution. . The random variable V with the p.d.f. [{B([alpha], [beta])}.sup.-1][v.sup.[alpha]-1][(1 + v).sup.-([alpha]+[beta])], v > 0 (1.2) where [alpha] > 0 and [beta] > 0, is said to have a beta type 2 distribution with parameters ([alpha], [beta]). Since (??) can be obtained from (??) by the transformation v = u/(1 - u) some authors call the distribution of v an inverted inverted reverse in position, direction or order. inverted L block a pattern of local filtration anesthesia commonly used in laparotomy in the ox. beta distribution. The inverted beta distribution arises from a linear transformation of the F distribution. The beta type 1 and beta type 2 are very flexible distributions for positive random variables and have wide applications in statistical analysis, e.g., see Johnson, Kotz and Balakrishnan [?]. Systematic treatment of matrix variate generalizations of the beta type 1 and the beta type 2 distributions is given in Gupta Gupta (g  p`tə), Indian dynasty, A.D. c.320–c.550, whose empire at its height encompassed much of N India. Ancient Indian culture reached a high point during this period. and Nagar
American music producer who founded Motown Records (1959) and inspired the "Motown sound," popular music heavily influenced by soul and rhythm and blues. [?], Ng and Kotz [?] and McDonald and Xu [?]. By using the transformation w = u/(2 - u), the beta type 3 density is obtained as (Gupta and Nagar [?]), [2.sup.[alpha]][{B([alpha], [beta])}.sup.-1][w.sup.[alpha]-1][(1 - w).sup.[beta]-1][(1 + w).sup.-([alpha]+[beta]), 0 < w < 1. (1.3) In this article we will study properties of the beta type 3 distribution and its multivariate generalization. In Section 2, several properties of the beta type 3 distribution including moment generating function, marginal distribution In probability theory, given two jointly distributed random variables X and Y, the marginal distribution of X is simply the probability distribution of X ignoring information about Y and moments are studied. The results such as the cumulative distribution function, the mean and the variance The discrepancy between what a party to a lawsuit alleges will be proved in pleadings and what the party actually proves at trial. In Zoning law, an official permit to use property in a manner that departs from the way in which other property in the same locality of the beta type 3 random variable involve Gauss hypergeometric function which is easily computable computable - computability theory using the software Mathematica Mathematical software for the Macintosh, DOS, Windows, OS/2 and various Unix platforms from Wolfram Research, Inc., Champaign, IL (www.wolfram.com). Launched in 1988, Mathematica includes numerical, graphical and symbolic computation capabilities, all linked to the Mathematica programming . We define the multivariate generalization of the beta type 3 distribution in Section 3. It may be noted here that the beta type 3 distribution too is defined on the finite finite - compact interval and may serve as an alternative to the beta type 1 distribution for many practical applications. The multivariate generalization of the beta type 3 distribution is a new member of the multivariate Liouville family ([?], [?]) and deserves to be considered for further research. 2. Properties In this section we study properties of the random variable distributed as univariate beta type 3. If a random variables U has the p.d.f (??), then we will write U ~ B1([alpha], [beta]), and if the p.d.f. of a random variable V is given by (??), then V ~ B2([alpha], [beta]). The density (??) will be designated by W ~ B3([alpha], [beta]). From the density of the beta type 3 variable it can be shown that [[integral].sup.1.sub.0][w.sup.[alpha]-1](1 - w).sup.[beta]-1][(1 + w).sup.-([alpha]+[beta])] dw = [2.sup.-[alpha]]B([alpha], [beta]). (2.1) The complementary cumulative distribution function of W is obtained as [MATHEMATICAL EXPRESSION A group of characters or symbols representing a quantity or an operation. See arithmetic expression. NOT REPRODUCIBLE re·pro·duce v. re·pro·duced, re·pro·duc·ing, re·pro·duc·es v.tr. 1. To produce a counterpart, image, or copy of. 2. Biology To generate (offspring) by sexual or asexual means. IN ASCII ASCII or American Standard Code for Information Interchange, a set of codes used to represent letters, numbers, a few symbols, and control characters. Originally designed for teletype operations, it has found wide application in computers. .] (2.2) where the second step has been obtained by substituting t = (1 - u)/(1 + u). Further substitution Substitution Arsinoë put her own son in place of Orestes; her son was killed and Orestes was saved. [Gk. Myth.: Zimmerman, 32] Barabbas robber freed in Christ’s stead. [N.T.: Matthew 27:15–18; Swed. Lit. of z = (1 + w)u/(1 - w) in (??) yields [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.3) where the last line has been obtained by using an integral representation of the Gauss' hypergeometric function (Luke Luke early Christian; the “beloved physician.” [N.T.: Luke] See : Evangelism [?]). In the following theorem theorem, in mathematics and logic, statement in words or symbols that can be established by means of deductive logic; it differs from an axiom in that a proof is required for its acceptance. , using series expansion of [(1 + w).sup.-([alpha]+[beta])] in the p.d.f. (??), we obtain another form of representation of the beta type 3 density. Theorem 2.1. Let W ~ B3([alpha], [beta]), then the p.d.f. of W can be expressed as [[infinity infinity, in mathematics, that which is not finite. A sequence of numbers, a1, a2, a3, … , is said to "approach infinity" if the numbers eventually become arbitrarily large, i.e. ].summation summation n. the final argument of an attorney at the close of a trial in which he/she attempts to convince the judge and/or jury of the virtues of the client's case. (See: closing argument) over (r=0)][[([beta]).sub.r]/[2.sup.[beta]+r]r!][{B([alpha], [beta] + r)}.sup.-1] [w.sup.[alpha]-1][(1 - w).sup.[beta]+r-1], 0 < w < 1 (2.4) That is, the p.d.f. of the beta type 3 variable is a mixture of beta type 1 densities. Proof. Substituting the series expansion of [(1 + w).sup.-([alpha]+[beta])] given by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.5) in (??) and simplifying, we obtain the desired result. Theorem 2.2. Let W ~ B3([alpha], [beta]). Then, the moment generating function (m.g.f.) [M.sub.W](t) of W is given by [M.sub.W](t) = [[infinity].summation over (r=0)][([beta]).sub.r]/ [2.sup.[beta]+r]r!][sub.1][F.sub.1]([alpha]; [alpha] + [beta] + r; t), t > 0 (2.6) where [sub.1][F.sub.1] is the confluent hypergeometric function In mathematics, there are two types of functions known as confluent hypergeometric functions. One is the family of solutions to a differential equation known as Kummer's equation; these are called Kummer's confluent hypergeometric function, or simply Kummer's function (Abramowitz and Stegun Abramowitz and Stegun is the informal name of a mathematical reference work edited by Milton Abramowitz and Irene Stegun of the U.S. National Bureau of Standards (now the National Institute of Standards and Technology). [?]). Proof. The proof is immediate from (??) upon realizing that the m.g.f. of a beta type 1 distribution with parameters ([alpha], [beta] + r) is [sub.1][F.sub.1]([alpha]; [alpha] + [beta] + r; t). The relationship between beta type 1, type 2 and type 3 random variables is exhibited in the following theorem. The proof is straightforward and is left to the reader. Theorem 2.3. Let U ~ B1([alpha], [beta]), V ~ B2([alpha], [beta]) and W ~ B3([alpha], [beta]). Then, (i) [(1 + U).sup.-1](1 - U) ~ B3([beta], [alpha]), (ii) 2W/(1 + W) ~ B1([alpha], [beta]) (iii) [(1 + W).sup.-1](1 - W) ~ B1([beta], [alpha]), (iv) V/(2 + V ) ~ B3([alpha], [beta]), (v) [(1 + 2V).sup.-1] ~ B3([beta], [alpha]), (vi) 2W/(1 - W) ~ B2([alpha], [beta]), and (vii) (1 - W)[W.sup.-1]/2 ~ B2([beta], [alpha]). A random variable Y is said to have a gamma distribution with parameters [kappa Kappa Used in regression analysis, Kappa represents the ratio of the dollar price change in the price of an option to a 1% change in the expected price volatility. Notes: Remember, the price of the option increases simultaneously with the volatility. ] (> 0) and [theta Theta A measure of the rate of decline in the value of an option due to the passage of time. Theta can also be referred to as the time decay on the value of an option. If everything is held constant, then the option will lose value as time moves closer to the maturity of the option. ] (> 0), denoted as Y ~ Ga([kappa], [theta]), if its p.d.f. is given by [{[[theta].sup.[kappa]][GAMMA]([kappa])}.sup.-1]exp exp abbr. 1. exponent 2. exponential (-y/[theta]) [y.sup.[kappa]-1], y > 0. Let [Y.sub.1] ~ Ga([alpha], [theta]) and [Y.sub.2] ~ Ga([beta], [theta]) be independent. Then, it is well known that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.7) where X [??] Z means that X and Z have identical distribution. Now, it is easy to see that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.8) Further, using the stochastic By guesswork; by chance; using or containing random values. stochastic - probabilistic representations (??) and (??) all the results of the Theorem ?? can be established easily. The representation (??) suggests the obvious extension [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.9) where V ~ B2([alpha], [beta]). The p.d.f. of [W.sub.c] is [c.sup.[alpha]][{B([alpha], [beta])}.sup.-1][w.sup.[alpha]-1] [(1 - w).sup.[beta]-1][[1 + (c - 1)w].sup.-([alpha]+[beta])], 0 < w < 1. (2.10) Further, using (??) it is easy to see that for [alpha] > 0, [beta] > 0 and K > 0, [[integral].sup.1.sub.0][w.sup.[alpha]-1][(1 - w).sup.[beta]-1] [(1 + Kw).sup.-([alpha]+[beta])] dw = [(1 + K).sup.-[alpha]]B([alpha], [beta]). (2.11) For K = 1, the above integral reduces to (??). Theorem 2.4. Let W ~ B3([alpha], [beta]), then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.12) where Re(r + [alpha]) > 0, Re(s + [beta]) > 0 and [sub.2][F.sub.1] is the Gauss hypergeometric function. Proof. From the density of W, we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] Writing [(1 + w).sup.-([alpha]+[beta]+t)] = [2.sup.-([alpha]+[beta]+t)] [[1 - (1 - w)/2].sup.-([alpha]+[beta]+t)] and substituting z = 1 - w, we obtain [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] Finally, using the integral representation of Gauss hypergeometric function (Abramowitz and Stegun[?], Luke [?]), we get the desired result. Substituting r = t = h and s = 0 in (??) and using the result [sub.2] [F.sub.1](beta], [alpha] + [beta] + h; [alpha] + [beta] + h; 1/2) = [2.sup.-[beta]], we get E[[W.sup.h]/[(1 + W).sup.h]] = [[GAMMA]([alpha] + [beta])[GAMMA]([alpha] + h)]/[[2.sup.h][GAMMA]([alpha])[GAMMA]([alpha] + [beta] + h)], Re(h + [alpha]) > 0. The above expression can also be obtained by observing that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.13) where U ~ B1([alpha], [beta]). The mean and the variance of the beta type 3 distribution can be obtained as particular cases of the results derived above. Substituting s = t = 0 and r = 1, 2 in the Theorem ?? and using the definitions of mean and variance, we obtain E(W) = [[alpha]/[2.sup.[beta]]([alpha] + [beta])][sub.2][F.sub.1] ([beta], [alpha] + [beta]; [alpha] + [beta] + 1; 1/2) and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] If U ~ B1([alpha], [beta]), then E(U) = [alpha]/[[alpha] + [beta]] and Var(U) = [alpha][beta]/[[([alpha] + [beta]).sup.2]([alpha] + [beta] + 1)]]. Now, using the result [sub.2][F.sub.1] ([beta], [alpha] + [beta]; [alpha] + [beta] + 1; 1/2) [less than or equal to] [2.sup.[beta]], it is straightforward to show that E(W) [less than or equal to] E(U) for all [alpha] and [beta]. For variances, such an inequality inequality, in mathematics, statement that a mathematical expression is less than or greater than some other expression; an inequality is not as specific as an equation, but it does contain information about the expressions involved. does not seem possible. The Gauss hypergeometric functions [sub.2][F.sub.1](a, b; c; x) can be computed using Mathematica by providing values of a, b, c and x. As we have already shown above, Var(W) involves [sub.2][F.sub.1] (beta], [alpha] + [beta]; [alpha] + [beta] + 1; 1/2) and [sub.2][F.sub.1]([beta], [alpha] + [beta]; [alpha] + [beta] + 2; 1/2) which come in Mathematica as Hypergeometric Hypergeometric can refer to various related mathematical topics:
3. Dirichlet Type 3 Distribution The multivariate generalization of the beta type 1 density is defined by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.1) where [[alpha].sub.i] > 0, i = 1, ..., n + 1. This distribution has been considered by several authors and is well known in the scientific literature as the Dirichlet distribution In probability and statistics, the Dirichlet distribution (after Johann Peter Gustav Lejeune Dirichlet), often denoted Dir(α), is a family of continuous multivariate probability distributions parametrized by the vector α of positive reals. . To distinguish it from the distribution that we will define shortly, we will write ([U.sub.1], ..., [U.sub.n]) ~ D1([[alpha].sub.1], ..., [[alpha].sub.n]; [[alpha].sub.n+1]) if the joint density of [U.sub.1], ..., [U.sub.n] is given by (??). The positive random variables [V.sub.1], ..., [V.sub.n] are said to have the Dirichlet (multivariate beta) type 2 distribution, denoted as ([V.sub.1], ..., [V.sub.n]) ~ D2([[alpha].sub.1], ..., [[alpha].sub.n]; [[alpha].sub.n+1]), if their joint p.d.f. is given by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.2) where [[alpha].sub.i] > 0, i = 1, ..., n + 1. A natural multivariate generalization of the beta type 3 distribution can be given as follows. Definition 3.1. The positive random variables [W.sub.1], ..., [W.sub.n] are said to have the Dirichlet (multivariate beta) type 3 distribution, denoted as ([W.sub.1], ..., [W.sub.n]) ~ D3([[alpha].sub.1], ..., [[alpha].sub.n]; [beta]), if their joint p.d.f. is given by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.3) where C([[alpha].sub.1], ..., [[alpha].sub.n]; [beta]) is the normalizing constant The concept of a normalizing constant arises in probability theory and a variety of other areas of mathematics. Definition and examples In probability theory, a normalizing constant . The Dirichlet type 1 and Dirichlet type 3 distributions belong to the Liouville family of distributions defined by (Gupta and Song [?]), [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.4) where [[alpha].sub.i] > 0, i = 1, ..., n and g is a non-negative real valued function defined on the interval (0, 1) such that the integral [[integral].sup.1.sub.0] [t.sup.r-1]g(t) dt exists for all r > 0. This distribution will be denoted by ([W.sub.1], ..., [W.sub.n]) ~ [[??].sup.(2).sub.n]([[alpha].sub.1], ..., [[alpha].sub.n]; g). The normalizing constant of the density (??) depends on the function g and can be evaluated explicitly. In general, this constant can be written in terms of Weyl fractional fractional size expressed as a relative part of a unit. fractional catabolic rate the percentage of an available pool of body component, e.g. protein, iron, which is replaced, transferred or lost per unit of time. integral. The normalizing constant in (??) is given by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] where the last line has been obtained by using Liouville-Dirichlet integral. Now, evaluating the above integral using (??) and simplifying the result, we get [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.5) The next theorem derives the Dirichlet type 3 distribution from the Dirichlet type 1 distribution. Theorem 3.1. Let ([U.sub.1], ... ,[U.sub.n]) ~ D1([[alpha].sub.1], ..., [[alpha].sub.n]; [[alpha].sub.n+1]). Define [W.sub.i] = [U.sub.i]/(1 - 2[[summation].sup.n.sub.i=1] [[U.sub.i]), i = 1, ..., n. Then, ([W.sub.1], ..., [W.sub.n]) ~ D3 ([[alpha].sub.1], ..., [[alpha].sub.n]; [[alpha].sub.n+1]). Proof. Substituting [u.sub.i] = 2[w.sub.i]/(1 + [[summation].sup.n.sub.i=1][w.sub.i]), i = 1, ..., n with the Jacobian For the French Revolution faction, see Jacobin. For the followers of James II of England and VII of Scotland, see Jacobitism. For other uses see Jacobean. In vector calculus, the Jacobian is shorthand for either the Jacobian matrix or its determinant, the of transformation J([u.sub.1], ..., [u.sub.n] [right arrow] [w.sub.1], ..., [w.sub.n]) = [2.sup.n][(1 + [[summation].sup.n.sub.i=1][w.sub.i]).sup.-(n+1)] in (??) and simplifying, we get the desired result. Let [Y.sub.1], ..., [Y.sub.n+1] be independent random variables, [Y.sub.i] ~ Ga([[alpha].sub.i], [theta]), i = 1, ..., n + 1. Then, it is well known that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.6) and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.7) Now, using (??) and Theorem ??, it is easy to see that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.8) Further, from (??) and (??), it follows that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] If ([U.sub.1], ..., [U.sub.n]) ~ D1([[alpha].sub.1], ..., [[alpha].sub.n]; [[alpha].sub.n+1]), then it is well known that for 1 [less than or equal to] m [less than or equal to] n, ([U.sub.1], ..., [U.sub.m]) ~ D1([[alpha].sub.1], ..., [[alpha].sub.m]; [[summation].sip.n+1].sub.i=m+1] [[alpha].sub.i]). In the following theorem we will derive similar result for the Dirichlet type 3 variables. However, the marginal distribution in this case is not a Dirichlet type 3 distribution. Theorem 3.2. Let ([W.sub.1], ..., [W.sub.n]) ~ D3([[alpha].sub.1], ..., [[alpha].sub.n]; [beta]). Then, the marginal p.d.f. of ([W.sub.1], ..., [W.sub.m]) is given by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.9) where 0 < [w.sub.i] < 1, i = 1, ..., m and [[summation].sup.m.sub.i=1] [w.sub.i] < 1. Proof. The marginal density of [W.sub.1], ..., [W.sub.m], m [less than or equal to] n can be obtained by integrating out [w.sub.m+1], ..., [w.sub.n]. Substituting [x.sub.n] = [(1 - [[summation].sup.n-1.sub.i=1]).sup.-1][w.sub.n] with the Jacobian J([w.sub.n] [right arrow] [x.sub.n]) = (1 - [[summation].sup.n-1.sub.i=1][w.sub.i], in (??), the joint density of [W.sub.1], ..., [W.sub.n-1] and [X.sub.n] is derived as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] Now, integrating this expression with respect to [x.sub.n] we find the marginal density of [W.sub.1], ..., [W.sub.n-1] is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] where 0 < [w.sub.i] < 1, i = 1, ..., n - 1, [[summation].sup.n-1.sub.i=1][w.sub.i] < 1. Using the result, [sub.2][F.sub.1](a, b; c; x) = [(1 - x).sup.-b] [sub.2][F.sub.1](c - a, b; c; -x[(1 - x).sup.-1]) (3.10) the Gauss hypergeometric function given in the above expression can be re-written as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] Hence, the joint marginal density of [W.sub.1], ..., [W.sub.n-1] is rewritten as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.11) Now, we consider the transformation [x.sub.n-1] = [(1 - [[summation].sup.n-2.sub.i=1][w.sub.i]).sup.-1][w.sub.n-1]. Then, the joint density of [W.sub.1], ..., [W.sub.n-2] and [X.sub.n-1] is derived as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] The joint p.d.f. of [W.sub.1], ..., [W.sub.n-2] is obtained by integrating the above expression with respect to [x.sub.n-1]. Substituting [y.sub.n-1] = 1 - [x.sub.n-1] and integrating out [y.sub.n], we obtain the joint p.d.f. of [W.sub.1], ..., [W.sub.n-2] as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.12) By examining the form of the density functions in (??) and (??) it can be proved in an inductive inductive 1. eliciting a reaction within an organism. 2. inductive heating a form of radiofrequency hyperthermia that selectively heats muscle, blood and proteinaceous tissue, sparing fat and air-containing tissues. manner that the marginal density of [W.sub.1], ..., [W.sub.m] is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] Finally, applying (??) to the Gauss hypergeometric function of this last expression, we obtain the desired result. It can clearly be observed that the p.d.f. in (??) is not a Dirichlet type 3 density and differs by a factor involving [sub.2][F.sub.1]. Theorem 3.3. Let ([W.sub.1], ..., [W.sub.n]) ~ D3([[alpha].sub.1], ..., [[alpha].sub.n], [beta]) and define the random variables [X.sub.s+1], ..., [X.sub.n] as [X.sub.i] = [(1 - [[summation.sup.s.sub.i=1][W.sub.i]).sup.-1][W.sub.i], i = s + 1, ..., n. Then ([X.sub.s+1], ..., [X.sub.n]) ~ D3([[alpha].sub.s+1], ..., [[alpha].sub.n], [beta]). Proof. Transforming [x.sub.i] = [(1 - [[summation].sup.s.sub.i=1][w.sub.i].sup.-1][w.sub.i], i = s + 1, ..., n with the Jacobian J([w.sub.s+1], ..., [w.sub.n] [right arrow] [x.sub.s+1], ..., [x.sub.n]) = [(1 - [[summation].sup.s.sub.i=1][w.sub.i]).sup.n-s] in (??) and integrating [w.sub.1], ..., [w.sub.s], we get the joint density of [X.sub.s+1], ..., [X.sub.n] as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.13) where 0 < [x.sub.i] < 1, i = 1, ..., s, [[summation].sup.s.sub.i=1] [x.sub.i] < 1 Now, using Liouville-Dirichlet integral and (??), the integral given in (??) is evaluated as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.14) Finally, using (??) in (??) we get the desired result. In the next theorem we give distribution of partial sums of random variables distributed jointly as Liouville type 2. Theorem 3.4. Let ([W.sub.1], ..., [W.sub.n]) ~ [[??].sup.(2).sub.n]([[alpha].sub.1], ..., [[alpha].sub.n]; g) and [n.sub.1], ..., [n.sub.l] be positive integers such that [[summation].sup.l.sub.i=1] [n.sub.i] = n. Further, let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], [n.sup.*.sub.0] = 0, [n.sup.*.sub.i] = [[summation.sup.i.sub.j=1] [n.sub.j], i = 1, ..., l. Define [Z.sub.j] = [W.sub.j]/[W.sub.(i)], j = [n.sup.*.sub.i-1] + 1, ..., [n.sup.*.sub.i] - 1 and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], i = 1, ..., l. Then, (i) ([W.sub.(1)], ..., [W.sub.(l)]) and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], i = 1, ..., l, are independently distributed, (ii) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], i = 1, ..., l, and (iii) ([W.sub.(1)], ..., [W.sub.(l)] ~ [[??].sup.(2).sub.l]([[alpha].sub.(1)], ..., [[alpha].sub.(l); g). Proof. Making the transformation [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [z.sub.j] = [w.sub.j]/[w.sub.(i)], j = [n.sup.*.sub.i-1] + 1, ..., [n.sup.*.sub.i] - 1, i = 1, ..., l with the Jacobian [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.15) in the joint density of ([W.sub.1], ..., [W.sub.n]) given by (??), we get the joint density of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], [W.sub.(i)], i = 1, ..., l as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.16) where 0 < [w.sub.(i)] < 1, i = 1, ..., l, [[summation].sup.l.sub.i=1][w.sub.(i)] < 1, 0 < [z.sub.j] < 1, j = [n.sup.*.sub.i-1] + 1, ..., [n.sup.*.sub.i] - 1, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], i = 1, ..., l. From the factorization fac·tor·ize tr.v. fac·tor·ized, fac·tor·iz·ing, fac·tor·iz·es Mathematics To factor. fac in (??), it is easy to see that ([W.sub.(1)], ..., [W.sub.(l)]) and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], i = 1, ..., l, are independently distributed. Further [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], i = 1, ..., l and ([W.sub.(1)], ..., [W.sub.(l)]) ~ [[??].sup.(2).sub.l]([[alpha].sub.(1)], ..., [[alpha].sub.(l)]; g). Corollary corollary: see theorem. 3.4.1. Let ([W.sub.1], ..., [W.sub.n]) ~ Dk([[alpha].sub.1], ..., [[alpha].sub.n]; [beta]), k = 1, 3. Then, (i) ([W.sub.(1)], ..., [W.sub.(l)]) and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], i = 1, ..., l, are independently distributed, (ii) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] i = 1, ..., l, (iii) ([W.sub.(1)], ..., [W.sub.(l)]) ~ D1([[alpha].sub.(1)], ..., [[alpha].sub.(l)]; [beta]) if ([W.sub.1], ..., [W.sub.n]) ~ D1([[alpha].sub.1], ..., [[alpha].sub.n]; [beta]), and ([W.sub.(1)], ..., [W.sub.(l)]) ~ D3([[alpha].sub.(1)], ..., [[alpha].sub.(l)]; [beta]) if ([W.sub.1], ..., [W.sub.n]) ~ D1([[alpha].sub.1], ..., [[alpha].sub.n]; [beta]), The joint moments of [W.sub.1], ..., [W.sub.n] are given by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.17) where W ~ B3([[summation].sup.n.sub.i=1][[alpha].sub.i], [beta]). Now, computing computing - computer [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] using Theorem ??, substituting for C([[alpha].sub.1], ..., [[alpha].sub.n], [beta]) and C([[summation].sup.n.sub.i=1][[alpha].sub.i], [beta]) from (??) and simplifying the resulting expression, we obtain [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] Now, we consider an approximation approximation /ap·prox·i·ma·tion/ (ah-prok?si-ma´shun) 1. the act or process of bringing into proximity or apposition. 2. a numerical value of limited accuracy. of the Dirichlet type 3 distribution when [beta] increases. Theorem 3.5. Let ([W.sub.1], ..., [W.sub.n]) ~ D3([[alpha].sub.1], ..., [[alpha].sub.n]; [beta]). If [beta] [right arrow] [infinity], then [beta]([W.sub.1], ..., [W.sub.n]) [??] ([X.sub.1], ..., [X.sub.n]) where the random variables [X.sub.1], ..., [X.sub.n] are distributed independently, [X.sub.i] ~ Ga([[alpha].sub.i], 1/2), i = 1, ..., n and "[??]" denotes the convergence in distribution. Proof. Consider the transformation [u.sub.i] = [beta][w.sub.i], i = 1, ..., n. Then, the joint p.d.f. of [U.sub.1], ..., [U.sub.n] is given by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] where [alpha] = [[summation].sup.n.sub.i=1][[alpha].sub.i]. Now, using the results [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] it is easy to see that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] where F([x.sub.1], ..., [x.sub.n]) is the joint cumulative distribution function (c.d.f.) of ([beta][W.sub.1], ..., [beta][W.sub.n]) and [F.sub.i]([x.sub.i]) is the c.d.f. of a gamma variate A gamma variate underlies a gamma distribution. Common applications of gamma variate functions are fitting of dilution of tracers, e.g. when recirculation occurs. with parameters ([[alpha].sub.i], 1/2), i = 1, ..., n. Acknowledgments See About this product. The research work of LC and DKN was supported by the Comite para el Desarrollo de la Investigacion, Universidad Universidad (English: University) may refer to:
Received May 5, 2004, Accepted June June: see month. 1, 2004. References [1] M. Abramowitz and I. A. Stegun, Handbook
This article is about reference works. For the subnotebook computer, see .
Dover (dō`vər), town (1991 pop. 33,461), Kent, SE England, on the Strait of Dover, beneath chalk cliffs (the "White Cliffs of Dover") c.375 ft (114 m) high. The small Dour River flows through the town. Publications, New York New York, state, United States New York, Middle Atlantic state of the United States. It is bordered by Vermont, Massachusetts, Connecticut, and the Atlantic Ocean (E), New Jersey and Pennsylvania (S), Lakes Erie and Ontario and the Canadian province of (1970). [2] N. Eugene, C. Lee and F. Famoye, Beta-normal distribution and its applications, Comm See comms. . Statist stat·ism n. The practice or doctrine of giving a centralized government control over economic planning and policy. stat  ist adj. .-Theory Methods, 31(2002), 497-512.
[3] Michael B. Gordy, Computationally com·pu·ta·tion n. 1. a. The act or process of computing. b. A method of computing. 2. The result of computing. 3. The act of operating a computer. convenient distributional assumptions for common-value auctions, Comput. Econom., 12(1998), 61-78. [4] A. K. Gupta and D. K. Nagar, Matrix Variate Distributions, Chapman & Hall/CRC, Boca Raton Boca Raton (bō`kə rətōn`), city (1990 pop. 61,492), Palm Beach co., SE Fla., on the Atlantic; inc. 1925. Boca Raton is a popular resort and retirement community that experienced significant industrial development in the 1970s and 80s. , 2000. [5] A. K. Gupta and D. K. Nagar, Matrix variate beta distribution., Int. J. Math. Math. Sci., 23 no. 7, (2000) 449-459. [6] A. K. Gupta and D. Song, Generalized gen·er·al·ized adj. 1. Involving an entire organ, as when an epileptic seizure involves all parts of the brain. 2. Not specifically adapted to a particular environment or function; not specialized. 3. Liouville distribution, Comput. Math. Appl., 32 no. 2 (1996), 103-109. [7] N. L. Johnson, S. Kotz and N. Balakrishnana, Continuous Univariate Distributions-2 , Second Edition, John Wiley John Wiley may refer to:
[8] Y. L. Luke, The Special Functions In mathematics, special functions are particular functions such as the trigonometric functions that have useful or attractive properties, and which occur in different applications often enough to warrant a name and attention of their own. and Their Approximations, Vol. 1 , Academic Press, New York, 1969. [9] J. B. McDonald and Yexiao J. Xu, A generalization of the beta distribution with applications, J. Econometrics econometrics, technique of economic analysis that expresses economic theory in terms of mathematical relationships and then tests it empirically through statistical research. , 66 (1995), 133-152. Beta Type 3 Distribution and Its Multivariate Generalization 241 [10] K. W. Ng and S. Kotz, Kummer-Gamma and Kummer-Beta univariate and multivariate distributions. Research Report No. 84 (1995), Department of Statistics, The University of Hong Kong The University of Hong Kong (commonly abbreviated as HKU, pronounced as "Hong Kong U") is the oldest tertiary institution in Hong Kong. Its motto is "Sapientia et Virtus" in Latin, and " , Hong Kong Hong Kong (hŏng kŏng), Mandarin Xianggang, special administrative region of China, formerly a British crown colony (2005 est. pop. 6,899,000), land area 422 sq mi (1,092 sq km), adjacent to Guangdong prov. . [11] D. Song and A. K. Gupta, Properties of generalized Liouville distribution, Random Oper. Stochastic Equations, 5 no. 4 (1997), 337-348. * Mathematics Subject Classification. Primary 62H10; Secondary 62E15. Liliam Cardeno, Daya K. Nagar, and Luz Estela Sanchez Departamento de Matematicas, Universidad de Antioquia, Medellin, A. A. 1226, Colombia Table 1: Ratios of variances of thr beta type 3 distribution with the beta type 1 distribution [alpha] [beta] = 0.1 [beta] = 0.3 [beta] = 0.5 0.1 0.9975 0.884 0.7965 0.2 1.057 0.9354 0.8418 0.3 1.113 0.9852 0.8859 0.4 1.167 1.033 0.9288 0.5 1.219 1.08 0.9706 0.6 1.269 1.125 1.011 0.7 1.317 1.168 1.051 0.8 1.363 1.210 1.089 0.9 1.407 1.251 1.127 1 1.450 1.290 1.163 2 1.804 1.627 1.481 3 2.066 1.886 1.733 4 2.269 2.092 1.938 5 2.432 2.259 2.107 6 2.566 2.399 2.250 7 2.678 2.517 2.373 8 2.773 2.619 2.479 9 2.855 2.707 2.572 10 2.927 2.784 2.654 [alpha] [beta] = 0.7 [beta] = 0.9 [beta] = 1 0.1 0.7276 0.6723 0.6486 0.2 0.7679 0.7085 0.683 0.3 0.8073 0.744 0.7168 0.4 0.8459 0.7789 0.7501 0.5 0.8837 0.8132 0.7828 0.6 0.9205 0.8468 0.815 0.7 0.9566 0.8799 0.8467 0.8 0.9919 0.9122 0.8777 0.9 1.026 0.944 0.9083 1 1.060 0.9752 0.9383 2 1.359 1.256 1.210 3 1.603 1.490 1.439 4 1.804 1.687 1.633 5 1.973 1.854 1.800 6 2.118 1.999 1.944 7 2.243 2.125 2.071 8 2.352 2.236 2.182 9 2.448 2.335 2.282 10 2.533 2.423 2.371 [alpha] [beta] = 1.5 [beta] = 2 [beta] = 3 0.1 0.5585 0.4992 0.4274 0.2 0.5858 0.5216 0.4437 0.3 0.6128 0.5439 0.4599 0.4 0.6396 0.566 0.4761 0.5 0.666 0.588 0.4922 0.6 0.6923 0.6099 0.5083 0.7 0.7182 0.6315 0.5244 0.8 0.7438 0.653 0.5404 0.9 0.769 0.6744 0.5563 1 0.794 0.6955 0.5722 2 1.027 0.8965 0.7268 3 1.231 1.078 0.8724 4 1.410 1.242 1.008 5 1.568 1.389 1.135 6 1.708 1.523 1.253 7 1.833 1.643 1.362 8 1.945 1.753 1.463 9 2.047 1.853 1.557 10 2.138 1.945 1.645 |
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