A class of power series q-distributions
Charalambos A. Charalambides
Department of Mathematics, University of Athens
ccharal@math.uoa.gr
Abstract
A class of power series q-distributions, generated by considering a q-
Taylor expansion of a parametric function into powers of the parameter,
is discussed. The q-Poisson (Heine and Euler), q-binomial, negative
q-binomial and q-logarithmic distributions belong in this class. The
probability generating functions and q-factorial moments of the power
series q-distributions are derived. In particular, the q-mean and the
q-variance are deduced.
1 Introduction
Benkherouf and Bather[BB88] derived the Heine and Euler distributions, which constitute q-analogs of the
Poisson distribution, as feasible priors in a simple Bayesian model for oil exploration. The probability function
of the q-Poisson distributions is given by (Charalambides[Cha16, p. 107])
λx
px (λ; q) = Eq (−λ) , x = 0, 1, . . . ,
[x]q !
where 0 < λ < 1/(1 − q) and Q∞ 0 < q < 1 (Euler distribution) or 0 < λ < ∞ and 1 < q < ∞ (Heine
i−1
distribution).
Q∞ Also, Eq (t) = i=1 (1 + t(1 − q)q ) is a q-exponential function. It should be noted that
i−1 −1
eq (t) = i=1 (1 − t(1 − q)q ) is another q-exponential function and that these q-exponential functions are
connected by Eq (t)eq (−t) = 1 and Eq−1 (t) = eq (t).
Kemp and Kemp [KK91], in their study of the Weldon’s classical dice data, introduced a q-binomial distribu-
tion. It is the distribution of the number of successes in a sequence of n independent Bernoulli trials, with the
odds of success at a trial varying geometrically with the number of trials. Kemp and Newton [KN90] further
studied it as stationary distribution of a birth and death process. The probability function of this q-binomial
distribution of the first kind is given by
x
θx q(2)
n
px (θ; q) = Qn , x = 0, 1, . . . , n,
x q i=1 (1 + θq i−1 )
where 0 < θ < ∞, and 0 < q < 1 or 1 < q < ∞.
Charalambides [Cha10] in his study of the q-Bernstein polynomials as a q-binomial distribution of the second
kind, introduced the negative q-binomial distribution of the second kind. It is the distribution of the number of
failures until the occurrence of the nth success in a sequence of independent Bernoulli trials, with the probability
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In: L. Ferrari, M. Vamvakari (eds.): Proceedings of the GASCom 2018 Workshop, Athens, Greece, 18–20 June 2018, published at
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1
of success at a trial varying geometrically with the number of successes. The probability function of this negative
q-binomial distribution of the second kind is given by
n
n+x−1 xY
px (θ; q) = θ (1 − θq i−1 ), x = 0, 1, . . . ,
x q i=1
where 0 < θ < 1 and 0 < q < 1.
A q-logarithmic distribution was studied by C. D. Kemp[Kem97] as a group size distribution. Its probability
function is given by
θx
px (θ; q) = [−lq (1 − θ)]−1 , x = 1, 2, . . . ,
[x]q
where 0 < θ < 1, 0 < q < 1, and
∞ ∞
!
Y 1 − θq x+i−1 X θj
−lq (1 − θ) = lim −1 =
x→0
i=1
1 − θq i−1 j=1
[j]q
is a q-logarithmic function.
The class of power series q-distributions, introduced in section 2, provides a unified approach to the study
of these distributions. Its probability generating function and q-factorial moments are derived. Demonstrating
this approach, the probability generating function and q-factorial moments of the q-Poisson (Heine and Euler),
q-binomial, negative q-binomial, and q-logarithmic distributions are obtained.
2 Power series q-distributions
Consider a positive function g(θ) of a positive parameter θ and assume that it is analytic with a q-Taylor
expansion
∞
X
g(θ) = ax,q θx , 0 < θ < ρ, ρ > 0, (1)
x=0
where the coefficient
1
ax,q = [Dx g(t)]t=0 ≥ 0, x = 0, 1, . . . , 0 < q < 1, or 1 < q < ∞, (2)
[x]q ! q
with Dq = dq /dq t the q-derivative operator,
dq g(t) g(t) − g(qt)
Dq g(t) = = ,
dq t (1 − q)t
does not involve the parameter θ. Clearly, the function
ax,q θx
px (θ; q) = , x = 0, 1, . . . , (3)
g(θ)
with 0 < q < 1 or 1 < q < ∞, and 0 < θ < ρ, satisfies the properties of a probability (mass) function.
Definition 2.1. A family of discrete q-distributions px (θ; q), θ ∈ Θ, q ∈ Q, is said to be a class of power series
q-distributions, with parameters θ, q and series function g(θ) if it has the representation (3), with series function
satisfying condition (1).
Remark 2.2. The range of x in (3), as in the case of the (usual) power series distributions), may be reduced.
Thus, we may have ax,q > 0 for x ∈ T , with
T = {x0 , x0 + 1, . . . , x0 + x1 − 1}, x0 ≥ 0, x1 ≥ 1.
Moreover, note that the truncated versions of the a power series q-distribution are also power series q-distributions
in their own right.
2
P∞ x
The probability generating function P (t) = x=0 px (θ; q)t , on using (1) and (3), is readily deduced as
g(θt)
P (t) = . (4)
g(θ)
Clearly, the mth q-derivative, with respect to t, of the probability generating function is
∞
dm
q P (t)
X
= px (θ; q)[x]m,q tx−m .
dq tm x=m
Thus, the mth q-factorial moment of the power series q-distribution, on using (4), is obtained as
m
dq g(θt) θ m dm q g(θ)
1
E([X]m,q ) = · = · , m = 1, 2, . . . . (5)
g(θ) dq tm t=1 g(θ) dq θm
In particular the q-mean is given by
θ dq g(θ)
E([X]q ) = · . (6)
g(θ) dq θ
Also, on using the expression
V ([X]q ) = qE([X]2,q ) − E([X]q ) E([X]q ) − 1 , (7)
the q-variance is obtained as
qθ2 d2q g(θ)
θ dq g(θ) θ dq g(θ)
V ([X]q ) = · − · · − 1 . (8)
g(θ) dq θ2 g(θ) dq θ g(θ) dq θ
Example 2.3. q-Poisson distributions. These are power series q-distributions, with series function g(λ) =
eq (λ) = 1/Eq (−λ), where 0 < λ < 1/(1 − q) and 0 < q < 1 or 0 < λ < ∞ and 1 < q < ∞. Since Dq eq (t) = eq (t)
and eq (0) = 1, it follows from (2) that
1 1
ax,q = [Dx eq (t)]t=0 = , x = 0, 1, . . . ,
[x]q ! q [x]q !
Also, the probability generating function of the q-Poisson distributions, on using (4), is deduced as
eq (λt)
P (t) = = Eq (−λ)eq (λt).
eq (λ)
The q-factorial moments, by (5) and since Dqm eq (λ) = eq (λ), are readily deduced as
E([X]m,q ) = λm , m = 1, 2, . . . .
In particular, the q-mean is given by
E([X]q ) = λ.
Also, using (7), the q-variance is obtained as
V ([X]q ) = qλ2 − λ(λ − 1) = λ(1 + (q − 1)λ).
Example
Qn 2.4. q-Binomial distribution of the first kind. The series function of this distribution is g(θ) =
i−1
i=1 (1 + θq ), where 0 < θ < ∞ and 0 < q < 1 or 1 < q < ∞. Since
Qn Qn
(1 + θq i−1 ) − i=1 (1 + θq i )
Dq g(θ) = i=1
(1 − q)θ
Qn−1 n−1
[(1 + θ) − (1 + θq n )] i=1 (1 + θq i ) Y
= = [n]q (1 + (θq)q i−1 ),
(1 − q)θ i=1
3
it follows successively that
n−x n−x
x
(1 + (θq x )q i−1 ) = [n]x,q q (2)
Y Y
Dqx g(θ) = [n]x,q q 1+2+···+(x−1) (1 + (θq x )q i−1 ),
i=1 i=1
for x = 1, 2, . . . , n. Thus, by (2),
1 n (x2)
ax,q = [Dqx g(t)]t=0 = q , x = 0, 1, . . . , n.
[x]q ! x q
Also, the probability generating function of the q-binomial distribution of the first kind, on using (4), is deduced
as Qn
(1 + θtq i−1 )
P (t) = Qi=1
n i−1 )
.
i=1 (1 + θq
The q-factorial moments, by (5) and since
n−m n
m m
Dqm g(θ) = [n]m,q q ( 2 ) (1 + (θq m )q i−1 ) = [n]m,q q ( 2 )
Y Y
(1 + θq i−1 ),
i=1 i=m+1
are obtained as m
[n]m,q θm q ( 2 )
E([X]m,q ) = Qm i−1 )
, m = 1, 2, . . . .
i=1 (1 + θq
In particular, the q-mean is
[n]q θ
E([X]q ) = .
(1 + θ)
Also, using (7) and, subsequently, the expression q[n − 1]q = [n]q − 1, the q-variance is obtained as
[n]q [n − 1]q θ2 q 2
[n]q θ [n]q θ
V ([X]q ) = + 1−
(1 + θ)(1 + θq) 1+θ 1+θ
[n]q θ [n]q θ(q − 1)
= 1+ .
(1 + θ)(1 + θq) 1+θ
Example 2.5. Negative
Qn q-binomial distribution of the second kind. It is a power series q-distribution, with series
function g(θ) = i=1 (1 − θq i−1 )−1 , where 0 < θ < 1 and 0 < q < 1. Since
Qn Qn
(1 − θq i−1 )−1 − i=1 (1 − θq i )−1
Dq g(θ) = i=1
(1 − q)θ
Qn+1 n+1
[(1 − θq n ) − (1 − θ)] i=1 (1 − θq i−1 ) Y
= = [n]q (1 − θq i−1 ),
(1 − q)θ i=1
it follows successively that
n+x
Y n+x
Y
Dqx g(θ) = [n]q [n + 1]q · · · [n + x − 1]q (1 − θq i−1 ) = [n + x − 1]x,q (1 − θq i−1 ),
i=1 i=1
for x = 1, 2, . . . . Thus, by (2),
1 x n+x−1
ax,q = [D g(t)]t=0 = , x = 0, 1, . . . .
[x]q ! q x q
Also, the probability generating function of the negative q-binomial distribution of the second kind, on using (4),
is deduced as Qn
(1 − θtq i−1 )−1
P (t) = Qi=1
n i−1 )−1
.
i=1 (1 − θq
4
The q-factorial moments, by (5) and since
n+m
Y
Dqm g(θ) = [n + m − 1]m,q (1 − θq i−1 )−1
i=1
n
Y m
Y
= [n + m − 1]m,q (1 − θq i−1 )−1 (1 − θq n+i−1 )−1 ,
i=1 i=1
are obtained as
m
Y
E([X]m,q ) = [n + m − 1]m,q θm (1 − θq n+i−1 )−1 , m = 1, 2, . . . .
i=1
In particular, the q-expected value is
[n]q θ
E([X]q ) = .
1 − θq n
Also, using (7) and, subsequently, the expression [n + 1]q = [n]q + q n , the q-variance is successively obtained as
[n]q [n + 1]q θ2 q
[n]q θ [n]q θ
V ([X]q ) = + 1 −
(1 − θq n )(1 − θq n+1 ) 1 − θq n 1 − θq n
[n]q θ [n]q θ(q − 1)
= 1 + .
(1 − θq n )(1 − θq n+1 ) 1 − θq n
Example 2.6. q-Logarithmic distribution. The series function of this distribution is
∞
X θj
g(θ) = −lq (1 − θ) = , 0 < θ < 1, 0 < q < 1.
j=1
[j]q
Taking successively its q-derivatives,
∞ ∞
X X j−1
Dqx g(θ) = [j − 1]x−1,q θj−x = [x − 1]q ! θj−x ,
j=x j=x
j−x q
and using the negative q-binomial formula
∞ x
X x+k−1 k Y
θ = (1 − θq i−1 )−1 ,
k q i=1
k=0
we find
x
Y
Dqx g(θ) = [x − 1]q ! (1 − θq i−1 )−1 .
i=1
Thus, by (2),
1 1
ax,q = [Dx g(t)]t=0 = , x = 1, 2, . . . .
[x]q ! q [x]q
Also, the probability generating function of the q-logarithmic distribution, on using (4), is deduced as
−lq (1 − θt)
P (t) = .
−lq (1 − θ)
The q-factorial moments, by (5) and since
m
Y
Dqm g(θ) = [m − 1]q ! (1 − θq i−1 )−1 ,
i=1
are obtained as
[−lq (1 − θ)]−1 [m − 1]q !θm
E([X]m,q ) = Qm i−1 )
, m = 1, 2, . . . .
i=1 (1 − θq
5
In particular, the q-mean value is
[−lq (1 − θ)]−1 θ
E([X]q ) = .
1−θ
Also, using (7), the q-variance is obtained as
[−lq (1 − θ)]−1 θ2 q [−lq (1 − θ)]−1 θ [−lq (1 − θ)]−1 θ
V ([X]q ) = + 1−
(1 − θ)(1 − θq) 1−θ 1−θ
−1 −1
[−lq (1 − θ)] θ 1 [−lq (1 − θ)] θ
= − .
1−θ 1 − θq 1−θ
References
[BB88] L. Benkherouf, J. A. Bather. Oil exploration: sequential decisions in the face of uncertainty.Journal of
Applied Probability, 25:529-543, 1988.
[Cha10] Ch. A. Charalambides The q-Bernstein basis as a q-binomial distribution. Journal of Statististical
Planning and Inference, 140:2184-2190, 2010.
[Cha16] Ch. A. Charalambides. Discrete q-Distributions. John Wiley & Sons, Hoboken, New Jersey, 2016.
[Kem97] C. D. Kemp. A q-logarithmic distribution. In Balakrishnan, N. (Ed.) Advances in Combinatorial
Methods and Applications to Probability and Statistics, Birkhäuser, Boston, MA, pp. 465-470, 1997.
[KK91] A. Kemp, C D. Kemp. Weldon’s dice data revisited. American Statistician, 45:216-222, 1991.
[KN90] A. Kemp, J. Newton. Certain state-dependent processes for dichotomized parasite population. Journal
of Applied Probability, 27:251-258, 1990.
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