"Science Stays True Here" Journal of Mathematics and Statistical Science, Volume 2016, 655-674 | Science Signpost Publishing
Raw and Central Moments of a Variety of Generalized Beta-Binomial Distributions via Stirling Numbers
Rachel Ogal
School of Mathematics, CBPS College, University of Nairobi, P. O. Box 30197-00100, Nairobi, Kenya.
Abstract
In this article, we consider the construction of various compound Binomials by mixing the success
parameter with generalized Beta distributions within the unit interval. We then proceed to calculate the
moments of the derived mixtures using Stirling numbers of the second kind.
Keywords: Binomial mixtures, Central moments, Factorial moments, Generalized Beta distributions,
Raw moments, Stirling numbers.
1. Introduction
If we let a random variable X to represent the total number of successes in n Bernoulli experiments, then X is said to be Binomially distributed and has parameters n and ϑ . ϑ denotes the probability of success hence lies within the unit interval, 01<<ϑ , and can be taken to be a continuous random variable with density g()ϑ . This formulates the density fx() referred to as a Binomial mixture.
1 n x nx− fx()=∫ ϑ( 1 − ϑ) g () ϑϑ d (1) 0 x
The Beta distribution offers a tractable prior to the Binomial, and has been used in this sense (see, for instance, Skellam (1948); Ishii and Hayakawa (1960); Chatfield and Goodhardt (1970)). Various generalizations of the Beta distribution have also been developed and used as priors to the Binomial (see, for instance, Gerstenkorn (2004); Rodriguez-Avi et al. (2007)).
656 Raw and Central Moments of a Variety of Generalized Beta-Binomial Distributions via Stirling Numbers We aim to construct Binomial mixtures whose priors are generalizations of the Beta, and derive some of their properties using stirling numbers of the second kind.
2. Stirling Numbers of the Second Kind
Stirling numbers are close relatives of the Binomial coefficients and are found in two kinds: stirling numbers of the first kind, stirling numbers of the second kind. They are named after James Stirling (1692-1770).
Consider the descending factorial x()k = xx( − 1)( x − 2) ( x −+ k 1) , which can be identified as a polynomial of x of degree k . Multiplying it out, and rearranging the terms gives us the following formula (as appearing in Gould (1978))
k j x()k =∑ sk() , jx j=0 where k =0 ,, 1 and sk(), j denotes the stirling numbers of the first kind.
Inversely, the k th power of x may be written in terms of a polynomial of factorials of x of degree j and stirling numbers of the second kind, Sk(), j, (see Joarder and Mahmood (1997)).
k k x=∑ Sk() , jx()j (2) j=0 where k =0 ,, 1 2 , .
An inductive proof for 2 is given in Gould (1978). We concentrate on stirling numbers of the second kind, Sk(), j and use them to calculate raw moments, hence central moments of derived Binomial mixtures.
Raw and Central Moments of a Variety of Generalized Beta-Binomial Distributions via 657 Stirling Numbers
k\j 0 1 2 3 4 5 6 7 8 9 0 1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 2 0 1 1 0 0 0 0 0 0 0 3 0 1 3 1 0 0 0 0 0 0 4 0 1 7 6 1 0 0 0 0 0 5 0 1 15 25 10 1 0 0 0 0 6 0 1 31 90 65 15 1 0 0 0 7 0 1 63 301 350 140 21 1 0 0 8 0 1 127 966 1701 1050 266 28 1 0 9 0 1 255 3025 7770 6951 2646 462 36 1
Table 1. Stirling numbers of the second kind, Sk(), j
Table 1 illustrates the Sk(), j, up to k = 9 . It can be extended using simple algebraic manipulation
(as given by Joarder and Mahmood (1997)) where for instance, to calculate Sj(4, ) for varying values of
j we expand 2 for k = 4 getting
4 xSxSxSxSxSx=,(4 0)(0) +, (4 1)(1) +, (4 2)(2) +, (4 3)(3) +, (4 4) (4) =,+,S(41) xS (4 2) x2 −+, x S (4 3) x32 − 3 x + 2 x + 43 2 S(4, 4) xx −+ 6 11 xx − 6
We then equate the coefficients of xx,,23 x and x4 found on the RHS, to corresponding values on the LHS, giving us the following sets of equations S(4,= 4) 1 SS(4,− 3) 6 (4 ,= 4) 0
SS(4,− 2) 3 (4 ,+ 3) 11 S (4 ,= 4) 0 SS(4,− 1) (4 , 2) + 2 S (4 ,− 3) 6 S (4 , 4) = 0
Upon solving these we get SSS(4,=,,=,,= 4) 1 (4 3) 6 (4 2) 7 and S(4,= 1) 1 .
Another method would be to use the formula
658 Raw and Central Moments of a Variety of Generalized Beta-Binomial Distributions via Stirling Numbers
j 1 i r k Sk(,= j )∑ ( − 1) (r − i) (3) j! i=0 i for 0 ≤≤jk and j > 0 , as given by (Roberts 1984)
3. Factorial Moments
Let X be a Binomial mixture, then the jth factorial moment of X about the origin, EX()j is given by
n 1 n x nx− EX= x ϑ(1− ϑ) f () ϑϑ d ()jj∑ ∫0 () x=0 x
n 1 xn!! nx− = ϑx (1− ϑ) fd () ϑϑ ∫0 ∑ x=0 (x−− j )! n xx ! !
n 1 −nj nx− = nϑx (1− ϑ) fd () ϑϑ ∫0 ∑ ()j x=0 xj−
n 1 −nj nx− = nϑj ϑxj− (1− ϑ) fd () ϑϑ ∫0 ()j ∑ x=0 xj−
j = ϑ ()nEj (4)
4. Raw and Central Moments
The raw moments of a discrete random variable X are defined by the formula
n kk = = = mki∑ xPx() x i EX i=1 where k refers to the order of the moment. Consider 2, taking the expectation of both sides of the equation gives us
k k = , EX∑ SkjEX()()j (5) j=0 hence a general formula for calculating the raw moments of the discrete generalized Beta-binomial distributions.
Raw and Central Moments of a Variety of Generalized Beta-Binomial Distributions via 659 Stirling Numbers The moments about the mean, or central moments of univariate discrete distributions are defined by the formula
n r µri=−=∑ (x m1 )( Px x ) (6) i=1
(6) can be simplified using Binomial expansion to get the following relation n r µri=−=∑ (x m1 )( Px x ) i=1 rn r rj− j j =−=∑∑(xii ) ( 1) ( m1 ) PX ( X ) ji=01j = r r jj =∑ ( − 1) (mm1 )rj− ,≥ ( r 1) j=0 j
The second order central moment µ2 , is called the variance, commonly denoted by Var() X or
σ 2 . In this paper, we only focus on the variances of generated Binomial mixtures.
5. Application
5.1. Beta-binomial distribution
From 1, the probability function of a Beta-binomial distribution is given by
α −1 β −1 1 n x nx− ϑϑ(1− ) fx()=∫ ϑϑ( 1 − ) dϑ 0 x B()αβ,
n Bx()+,−+αβ n x = , x B()αβ,
ΓΓαβ (xn=,,,;,>; 01 αβ 0 B( αβ , ) = ) (7) Γ+αβ
5.1.1. Factorial Moments
From 4, the jth factorial moments of the Beta-binomial are given by
j = ϑ EX()jj n () E() where E()ϑ j refers to the jth raw moment of the Beta prior calculated as
660 Raw and Central Moments of a Variety of Generalized Beta-Binomial Distributions via Stirling Numbers
j+−α 1 β −1 1 ϑϑ(1− ) Ed()ϑϑj = ∫0 B()αβ,
Bj()+,αβ = B()αβ, Giving
Bj()+,αβ EX= n (8) ()jj () B()αβ,
5.1.2. Raw and Central Moments
To find the kth raw moment of the Beta-binomial distribution, we substitute 8 into 5, to give
k k = , EX∑ SkjEX()()j j=0
k Bj()+,αβ =∑ Sk() , jn()j (9) j=0 B()αβ,
The mean is then calculated as
1 = , E[ X] ∑ S(1 jEX ) ()j j=0
B(αβ+, 1) =n B()αβ,
The 2nd moment m2 , is
2 2 Bj()+,αβ = , E X∑ S(2 jn ) ()j j=0 B()αβ,
nB(αβ+, 1)B ( α+ 2) , β =1 +− (n 1) BB()αβ, (α+, 1) β
2 The variance mm21− is
2 nB(αβ+, 1) B ( α+ 2) , βnB ( αβ +, 1) Var() X =1 +− (n 1) − B()αβ, BB (1)α+, β () αβ,
nB(αβ+, 1)B ( α+ 2) , β nB ( αβ +, 1) =1 +− (n 1) − B()αβ, BB (1)α+, β () αβ ,
Raw and Central Moments of a Variety of Generalized Beta-Binomial Distributions via 661 Stirling Numbers 5.2. McDonald’s Generalized Beta-binomial distribution
The probability function of a McDonald’s Generalized Beta-binomial (MGBB) distribution is given by
β − αλ−1 λ 1 1 λϑ− ϑ n x nx− 1 fx()=∫ ϑϑ( 1 − ) dϑ 0 x B()αβ, Further evaluation gives us
n λ 1 β − x x+−αλ 1 nx− λ 1 fx()= ϑ(1−− ϑ) 1 ϑϑd B()αβ, ∫0
n ∞ x λ β −1 1 nx− = ( −− 1) kxϑ+αλ −+1 k λ (1 ϑϑ) d ∑ ∫0 B()αβ, k =0 k
∞ n β −1 k Bx( +αλ + k λ ,−+ n x 1) =−,λ∑ ( 1) xkk =0 B()αβ,
(xn= 0 ,, 1 , ;αβλ , , >0) (10) as found by Manoj et al. (2013). We seek to re-write 10 in such a way as to eliminate the infinite series that occurs within it, hence easen computation. This leads to what we call the Moments method of deriving Binomial mixtures, as given by Sivaganesan and Berger (1993).
nx− The Moments method further evaluates 1 by expanding the (1−ϑ) , where ()nx− is a positive integer, found within the integral to give
nx− n nx− k 1 fx()= (−1) ϑxk+ g () ϑϑ d ∑ ∫0 xkk =0
nx− n nx− k xk+ = ∑ (−1() E ϑ ) (11) xkk =0
In this definition, E()ϑ xk+ is the ( xk+ ) th moment of the mixing distribution g()ϑ .
For the McDonald’s Generalized Beta distribution, the ( xk+ ) th moment is
662 Raw and Central Moments of a Variety of Generalized Beta-Binomial Distributions via Stirling Numbers
1 β − j λ xk++αλ −1 λ 1 Ed()ϑ= ϑ1− ϑϑ B()αβ, ∫0 xk+ (12) B()+,αβ = λ B()αβ,
Substituting (12) into (11) gives us
xk+ nx− B()+,αβ n nx− k λ fx()= ∑ (−1) (13) xkk =0 B()αβ,
(13) is a second way of writing the MGBB distribution.
Figure 1. Probability function plots of the Beta-binomial (BB) for αβ=,=,=8 4n 10 and the MGBB for
αβ=,=84 and varying values of λ
Raw and Central Moments of a Variety of Generalized Beta-Binomial Distributions via 663 Stirling Numbers Figure 1 displays a graphical comparison of the BB with the MGBB distribution for varying values of λ . The MGBB distribution becomes more negatively skewed as λ increases in value. For λ =1 the MGBB reduces to the BB distribution.
5.2.1. Factorial Moments
The jth raw moment of the McDonald’s Generalized Beta distribution from (12) is
j B()+,αβ λ
B()αβ,
So that the jth factorial moments of the MGBB distribution are
j B()+,αβ λ EX= n (14) ()jj () B()αβ,
5.2.2. Raw and Central Moments
Using 14, the kth raw moment of 10 can be easily derived as
j k B()+,αβ k λ = , E X∑ Sk() jn()j (15) j=0 B()αβ,
1 B()+,αβ λ So that m1 is n B()αβ, and m2 is given by
12 BB()+,αβ ()+,αβ 2 λλ EX=, S(2 1) n +,S(2 2) n (1) BB()αβ,,(2) ()αβ 12 nB()+,αβ B ()+,αβ λλ = 1+− (n 1) BB()αβ,,()αβ
2 The variance, mm21− , is
664 Raw and Central Moments of a Variety of Generalized Beta-Binomial Distributions via Stirling Numbers
2 1 21 nB()+,αβ B ()+,αβB ()+,αβ λ λλ2 Var() X = 1+− (n 1) −n αβ,,1 αβ BB() B()+,αβ() λ
1 21 nB()+,αβ B ()()+,αβB +, αβ λ λλ = 1+− (nn 1) − αβ,,1 αβ BB()B()+,αβ () λ
5.3. Libby and Novick’s Generalized Beta-binomial distribution
The probability function of Libby and Novick’s generalized Beta (LNBB) distribution is given by
β −1 cααϑϑ−1(1− ) gc()ϑ = αβ+ , (0 <ϑ <;,,> 1 αβ 0) B()αβ, [1−− (1c )ϑ] (see Libby and Novick (1982); Nadarajah and Kotz (2007)) From 1, the probability function of Libby and Novick’s Generalized Beta-binomial (LNBB) distribution is given by
x+−αα1 nx−+β −1 1 n ϑϑ(1− ) c = ϑ fx() ∫ αβ+ d 0 x B()αβ, [1−− (1c )ϑ]
α x+−αα1 nx−+β −1 n c B()αβ+,− xn x + 1 ϑϑ(1− ) c = ϑ ∫ α +ta d x B()αβ, 0 B(αβϑ+,− xn x +)1 −( 1 − c) (16) n α αβ+,− + c B() xn x =21Fx( +,+ααβαβ + +n;1 − c ) , x B()αβ, α −1 β −1 1 ϑϑ(1− ) =,,,;,,>αβ ; αγα ,+ β = ϑ (x 01 nc021Fz ( ; ;) ∫ γ d) 0 Bz(αβ,− )1( ϑ)
(16) reduces to the standard Beta-binomial distribution 7, when c =1.
Raw and Central Moments of a Variety of Generalized Beta-Binomial Distributions via 665 Stirling Numbers
Figure 2. Probability function plots of the Beta-binomial (BB) for αβ=,=,=8 4n 10 and the Libby and
Novick Beta-binomial (LNBB) for αβ=,=84 and varying values of c
Figure 2 compares plots of the BB with the LNBB distribution for varying values of the additional parameter c . The LNBB is left skewed for 01<
5.3.1. Factorial Moments
The jth raw moment of the LNBB distribution is given by
ααj+−1 β −1 1 c ϑϑ(1− ) ϑϑj = Ed()∫ αβ+ 0 Bc(αβ, )1 −−( 1 ) ϑ cBα ()αβ+, j = Fj (+,+ααβαβ ; + +j ;1 − c ) B()αβ, 21
666 Raw and Central Moments of a Variety of Generalized Beta-Binomial Distributions via Stirling Numbers Thus the jth factorial moment of the LNBB distribution is
α n()j cB()αβ+, j EX= (17) ()j B(αβ, )21 F ( α +, j αβαβ + ; + + jc ;1 − )
5.3.2. Raw and Central Moments
From 5, the raw moments of 16 are given by
k k = , EX∑ SkjEX()()j j=0 (18) k n cBα ()αβ+, j ()j =∑ Sk() , j 21F(α+, j αβαβ + ; + +j ;1 − c ) j=0 B()αβ,
n cBα ()αβ+, j Thus its mean m , is ()j F (α+, j αβαβ + ; + + jc ;1 − ) . 1 B()αβ, 21
m2 is given by
α 2 cB(αβ+, 1) EX= S( 2 , 1) n F (α+, 1 αβαβ + ; + + 1;1 −c ) (1) B()αβ, 2 1 cBα (αβ+, 2) +Sn(2 , 2) F (α+, 2 αβαβ + ; + + 2;1 −c ) (2) B()αβ, 2 1
ncα =BF(1)(1α +, β α +, αβαβ + ; + + 1;1) −c B()αβ, 21 (1)(2)(2nB−α +, β F α +, αβαβ + ; + + 2;1) −c 1+ 21 BF(αβ+, 1)21(α+, 1 αβαβ + ; + + 1;1 −c
and the variance is
ncα Var() X= B (1)(1α +, β F α +, αβαβ + ; + + 1;1) −c B()αβ, 21 (1)(2)(2nB−α +, β F α +, αβαβ + ; + + 2;1) −c 1+−21 BF(1)(1α+, β21 α +, αβαβ + ; + + 1;1) −c 2 α αβ+, nc B( 1) 21Fc(α+, 1 αβαβ + ; + + 1;1 − B()αβ,
Further simplified to give
Raw and Central Moments of a Variety of Generalized Beta-Binomial Distributions via 667 Stirling Numbers
ncα Var() X= B (1)(1α +, β F α +, αβαβ + ; + + 1;1) −c B()αβ, 21 (1)(2)(2nB−α +, β F α +, αβαβ + ; + + 2;1) −c 1+ 21 BF(1)(1α+, β21 α +, αβαβ + ; + + 1;1) −c α αβ+, nc B( 1) − 21Fc(α+, 1 αβαβ + ; + + 1;1 − ) B()αβ,
5.4. Gauss Hypergeometric-Binomial distribution
The probability function of the Gauss Hypergeometric distribution as suggested by Armero and Bayarri (1994); Nadarajah and Kotz (2007) is as follows
β −1 ϑϑα −1(1− ) g()ϑ = , γ B(αβ, )[1+ zϑ] 21 Fz (αγα, ; +− β ;) (0<ϑ < 1 ; αβ , > 0 ; − ∞ < γ < ∞; |z |< 1 for convergence)
The probability function of the Gauss Hypergeometric-Binomial (GHB) is thus derived as
x+−α 1 nx−+β −1 1 n ϑϑ(1− ) = ϑ fx() ∫ γ d 0 x B(αβ, )1( + zF ϑ) ( αγα , ; +− β ; z ) 21 n B(α+,− xn x + β ) F ( α +, x γα ; + β + n ;) − z = 21 , x BF(αβ, )21 ( αγα , ; +− β ;) z
(xn= 0 , 1 , , ; αβ , > 0 ; −∞< γ <∞ ) (19)
(19) reduces to the standard Beta-binomial distribution when z or γ equals 0 .
For γαβ= + and zc=−−(1 ) , the GHB distribution simplifies to the Libby and Novick
Beta-binomial distribution 16 Figure 3 shows the graph of the Beta-binomial distribution compared to that of the GHB distribution for varying values of γ and z . As the value of γ increases from zero, the graph of the GHB becomes more positively skewed. The reverse happens as γ decreases from zero.
668 Raw and Central Moments of a Variety of Generalized Beta-Binomial Distributions via Stirling Numbers
Figure 3. Probability function plots of the BB and the GHB distributions (a) BB for αβ=,=,=8 4n 10 and
GHB for αβ=,=,=8 4nz 10 ,=. 0 7 and varying values of γ (b) BB for αβ=,=,=8 4n 10 and GHB
for αβ=,=,=8 4n 10 and varying values of γ and z
5.4.1. Factorial Moments
From 4, the jth factorial moments of the GHB distribution are given by
j = ϑ EX()jj n () E
We then evaluate the jth moment of the Gauss Hypergeometric distribution as follows
α +−j 1 β −1 1 j ϑϑ(1− ) Edϑϑ= ∫ γ 0 B(αβ, )1( + zF ϑ) ( αγα , ; +− β ; z ) 21 B(α+, j β ) F ( α +, j γα ; + β + jz ;) − = 21 BF(αβ, )21 ( αγα , ; +− β ;) z
Raw and Central Moments of a Variety of Generalized Beta-Binomial Distributions via 669 Stirling Numbers To give us B(α+, j β )21 F ( α +, j γα ; + β + jz ;) − EX= n (20) ()jj () BF(αβ, )21 ( αγα , ; +− β ;) z
5.4.2. Raw and Central Moments
From 5 the kth raw moment of (19) is
k k = , EX∑ SkjEX()()j j=0 (21) k B(α+, j β ) F ( α +, j γα ; + β + jz ;) − =Sk() , jn 21 ∑ ()j j=0 BF(αβ, )21 ( αγα , ; +− β ;) z
BF(1)(1;α+, β α +, γα + β + 1;) − z so that the mean m , is n 21 . 1 BF(αβ, )21 ( αγα , ; +− β ;) z
m2 is
2 BF(1)(1;α+, β21 α +, γα + β + 1;) − z EX= S(2 , 1) n BF(αβ, )21 ( αγα , ; +− β ;) z BF(2)(2;α+, β α +, γα + β + 2;) − z +,Sn(2 2) 21 (2) BF(αβ, )21 ( αγα , ; +− β ;) z
BF(1)(1;α+, β α +, γα + β + 1;) − z = n 21 [1+ BF(αβ, )21 ( αγα , ; +− β ;) z (nB− 1) (αβ +, 2 )21 F (α+,2 γα ; ++− β 2;z ) BF(1)(1;α+, β21 α +, γα + β + 1;) − z
2 The variance, mm21− is then
BF(1)(1;α+, β α +, γα + β + 1;) − z Var() X= n 21 [1+ BF(αβ, )21 ( αγα , ; +− β ;) z (1)(2)(2;nB−α +, β F α +, γα + β + 2;) −z 21 − BF(1)(1;α+, β21 α +, γα + β + 1;) − z 2 BF(1)(1;α+, β α +, γα + β + 1;) − z n 21 BF(αβ, )21 ( αγα , ; +− β ;) z
670 Raw and Central Moments of a Variety of Generalized Beta-Binomial Distributions via Stirling Numbers This is further simplified to give
BF(α+, 1)( β α +, 1; γα ++− β 1;)(z nB − 1)(2)α +, β F ( α +, 2; γα ++− β 2;)z Var() x= n 21 1+ 21 BF(αβ, )21 ( αγα , ; + β ;) − z B (α+, 1)( β21 F α +, 1; γα + β + 1;) −z
nB(1)(1;α+, β F α +, γα + β + 1;) −z − 21 BF(αβ, )21 ( αγα , ; +− β ;) z
5.5. Confluent Hypergeometric-Binomial distribution
Given the probability function of the confluent hypergeometric distribution as suggested in Gordy (1988); Nadarajah and Kotz (2007)
β −1 ϑϑα −−1(1− ) e ϑλ g()ϑ = ,<<;,>;−∞<<∞(0ϑ 1 αβ0 λ ) BF(αβ, )11 (; αα +− β ; λ )
We derive the Confluent hypergeometric-Binomial (CHB) distribution as
x+−α 1 nx−+β −1 1 n ϑϑ(1− ) −ϑλ fx()= e dϑ ∫0 x BF(α, β )(11 αα +− β λ ) x+−α 1 nx−+β −1 n Bx()+,−+αβ n x 1 ϑϑ(1− ) −ϑλ = edϑ ∫0 x BF(αβ, )11 (; αα + β ; − λ )Bx ( +,−+ α n x β) n Bx(+,−+α n x β )( F x + αα ; + β + n ;) − λ = 11 , x BF(αβ, )11 (; αα +− β ; λ )
(xn= 0 , 1 , , ; αβ , > 0 ; −∞< λ <∞) (22)
(22) reduces to the Beta-binomial when λ = 0 Figure 4 shows the graph of the Beta-binomial distribution compared to that of the CHB distribution for varying values of the additional parameter λ . As the value of λ increases from zero, the graph of the CHB becomes more negatively skewed. The reverse happens as λ decreases from zero.
Raw and Central Moments of a Variety of Generalized Beta-Binomial Distributions via 671 Stirling Numbers
Figure 4. Probability function plots of the BB for αβ=,=,=8 4n 10 and CHB for αβ=,=,=8 4n 10 and
varying values of λ
5.5.1. Factorial Moments
j j = ϑ ϑ The jth factorial moment of 22 is given by EX()jj n () E, where E is the jth moment of the confluent hypergeometric distribution, found as follows
α +−j 1 β −1 −ϑλ 1 j ϑϑ(1− ) e Edϑϑ= ∫0 BF(αβ, ) (; αα +− β ; λ ) 11 Bx(+,−+α n x β )( F α + j ; αβ + + j ;) − λ = 11 BF(αβ, )11 (; αα +− β ; λ )
so that the CHB’s jth factorial moment is
j nB()j (α+, j β )(1 F 1 α + j ; αβ + + j ;) − λ EX= (23) BF(αβ, )11 (; αα +− β ; λ )
672 Raw and Central Moments of a Variety of Generalized Beta-Binomial Distributions via Stirling Numbers 5.5.2. Raw and Central Moments
k k From 5, the kth raw moment of 22 is given by = , . EX∑ j=0 SkjEX()()j
Further evaluation gives us
k k nB()j (α+, j β )(1 F 1 α + j ; αβ + + j ;) − λ EX= Sk() , j (24) ∑ j=0 BF(αβ, )11 (; αα +− β ; λ )
So that the mean, m1 is
BF(α+, 1 β ) ( α + 1; αβ + + 1; − λ ) EX[ ] = S(1 , 1) n 11 (1) BF(αβ, ) (; αα +− β ; λ ) 11 nB(α+, 1 β ) F ( α + 1; αβ + + 1; − λ ) = 11 BF(αβ, )11 (; αα +− β ; λ )
m2 is
2 BF(α+, 1 β )11 ( α + 1; αβ + + 1; − λ ) EX= S(2,+ 1) BF(αβ, )11 (; αα +− β ; λ ) BF(α+, 2 β ) ( α + 2; αβ ++− 2; λ ) Sn(2, 2) 11 (2) BF(αβ, )11 (; αα +− β ; λ )
nB(α+, 1 β ) F ( α + 1; αβ + + 1; − λ ) = 11 BF(αβ, )11 (; αα +− β ; λ ) (n− 1) nB (α +, 2 β ) F ( α + 2; αβ ++− 2; λ) 1+ 11 BF(α+, 1 β )11 ( α + 1; αβ + + 1; − λ )
The variance is thus given by
nB(α+, 1 β ) F ( α + 1; αβ ++− 1; λ ) (n − 1) nB (α +, 2 β ) F ( α + 2; αβ ++− 2; λ ) Var() X = 11 1+ 11 BF(αβ, )11 ( ααβ ; + +− 1; λ ) BF(α+, 1 β )11 ( α + 1; αβ + +− 1; λ )
BF(α+, 1 β ) ( α + 1; αβ + + 1; − λ ) −n 11 BF(αβ, )11 ( αα ; + β +− 1; λ )
Raw and Central Moments of a Variety of Generalized Beta-Binomial Distributions via 673 Stirling Numbers
Acknowledgement
Special thanks to Prof. J. A. M. Ottieno for providing the ideas upon which this article is based; to the School of Mathematics, University of Nairobi, for availing the resources necessary for the completion of this work.
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Published: Volume 2016, Issue 11 / November 25, 2016