EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 17, No. 4, 2024, 3847-3855
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Degenerate Moments and Expectation of Monomials

Dae San Kim1, Taekyun Kim2, Wonjoo Kim3, Jongkyum Kwon4,∗

Hyunseok Lee2,∗

1 Department of Mathematics, Sogang University, Seoul 121-742, Republic of Kore
2 Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea
3 Department of Applied Mathematics, Kyung Hee University, Yongin-si 17104,
Republic of Korea
4 Department of Mathematics Education, Gyeongsang National University, Jinju, 52828,
Republic of Korea

Abstract. The aim of this paper is twofold. Firstly, we obtain expressions of the degenerate
moments of a discrete nonnegative integer-valued random variable. Secondly, we get an expression
for the expectation of any monomial in discrete nonnegative integer-valued random variables.

2020 Mathematics Subject Classifications: 60-08, 60E0

Key Words and Phrases: Degenerate moments, expectation of monomials, discrete nonnegative
integer-valued random variables

1. Introduction

Let X be a discrete nonnegative integer-valued random variable. Then the probability
mass function on X is defined by pX(x) = P{X = x}. Oftentimes, we omit X from pX(x)
and denote it simply by p(x). This convention applies to other similar situations. The
cumulative distribution function on X is given by: for any nonnegative integer a,

FX(a) = P{X ≤ a} =

a∑
x=0

p(x) =

a∑
x=0

P{X = x}, (see [8–12, 14–19, 22, 23]). (1)

Let g(x) be a real valued function. Then the expectation of g(X) is given by

E
[
g(X)

]
=

∞∑
x=0

g(x)pX(x) =
∞∑
x=0

g(x)P{X = x}, (see [5, 8–12, 14–19, 22, 23]). (2)

∗Corresponding author.
∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v17i4.5604

Email addresses: dskim@sogang.ac.kr (D. S. Kim), tkkim@kw.ac.kr (T. Kim),
wjookim@khu.ac.kr (W. Kim), mathkjk26@gnu.ac.kr (J. Kwon), luciasconstant@kw.ac.kr (H. Lee)

https://www.ejpam.com 3847 Copyright: © 2024 The Author(s). (CC BY-NC 4.0)



J. Kwon et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3847-3855 3848

The n-th moment of X is defined by

E
[
Xn

]
=

∞∑
k=0

knp(k) =

∞∑
k=0

knP{X = k}, (see [17–19, 22, 23]). (3)

The variance of X is given by

Var(X) = E
[(
X − E[X]

)2]
= E

[
X2

]
−
(
E[X]

)2
, (see [23]). (4)

Let X and Y be discrete nonnegative integer-valued random variables. Then the joint
probability mass function of X and Y is defined by

p(x, y) = P{X = x, Y = y}, (see [4, 23]). (5)

We note that

P{X = x|Y = y} =
P{X = x, Y = y}

P{Y = y}
, (see [23]). (6)

Thus, by (5) and (6), we get

p(x, y) = P{X = x|Y = y}P{Y = y}. (7)

Let pX(x) and pY (y) be respectively the probability mass function of X and that of Y .
Then we have

pX(x) =
∑
y

P{X = x, Y = y} =
∑
y

p(x, y),

pY (y) =
∑
x

P{X = x, Y = y} =
∑
x

p(x, y).
(8)

The joint cumulative distribution function of X and Y is defined by: for any nonneg-
ative integers a and b,

FX,Y (a, b) = P{X ≤ a, Y ≤ b} =

b∑
y=0

a∑
x=0

p(x, y). (9)

By (7), we get

FX(a) = P{X ≤ a} = P{X ≤ a, Y ≤ ∞} = FX,Y (a,∞),

FY (b) = P{Y ≤ b} = P{X ≤ ∞, Y ≤ b} = FX,Y (∞, b).
(10)

For any λ ∈ R, the degenerate falling factorial sequence is defined by (see [8, 12, 14,
15, 17, 18, 21, 27])

(x)0,λ = 1, (x)n,λ = x(x− λ)(x− 2λ) · · ·
(
x− (n− 1)λ

)
, (n ≥ 1). (11)

With the notation in (11), we note that the degenerate exponentials are given by

exλ(t) =

∞∑
n=0

(x)n,λ
tn

n!
, (see [12, 14–18]). (12)



J. Kwon et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3847-3855 3849

We see that
lim
λ→0

(x)n,λ = xn, lim
λ→0

exλ(t) = ext.

The generating function of the degenerate moments E
[
(X)n,λ

]
of the random variable X

is given by

E
[
eXλ (t)

]
=

∞∑
n=0

E
[
(X)n,λ

] tn
n!
, (see [12, 14–18]).

In Section 1, we recall some necessary facts that are needed throughout this paper.
Section 2 contains the main results of this paper. Let X be a discrete nonnegative integer-
valued random variable. Then we obtain expressions for the r-th degenerate moment
E
[
(X)r,λ

]
(see (11)) as infinite series involving the cumulative distribution function FX

(see (1)) in Theorems 2.1 and 2.3. Assume that X,Y are discrete nonnegative integer-
valued random variables. In Theorem 2.2, we show that E[XY ] is equal to the double sum
over x, y of T (x, y), where T (x, y) = P{X > x, Y > y}. In Theorem 2.4, this is generalized
to the case of E[Xr1Y r2 ], where r1, r2 are any positive integers. Let r1, r2, · · · , rk be
positive integers, and let X1, X2, . . . , Xk be discrete nonnegative integer-valued random
variables. In Theorem 2.5, we get an expression for E[Xr1

1 Xr2
2 · · ·Xrk

k ] as a multiple sum
over x1, x2, . . . , xk, which involves T (x1, x2, . . . , xk). Here T (x1, x2, . . . , xk) = P{X1 >
x1, X2 > x2, . . . , Xk > xk}.

2. Degenerate moments and expectation of monomials

For r ∈ N, the r-th degenerate moment of X is given by

E
[
(X)r,λ

]
=

∞∑
x=0

pX(x)(x)r,λ =
∞∑
x=1

pX(x)(x)r,λ =
∞∑
x=0

pX(x+ 1)(x+ 1)r,λ (13)

=

∞∑
x=0

pX(x+ 1)

x∑
i=0

(
(i+ 1)r,λ − (i)r,λ

)
=

∞∑
i=0

(
(i+ 1)r,λ − (i)r,λ

) ∞∑
x=i

pX(x+ 1)

=

∞∑
i=0

(
(i+ 1)r,λ − (i)r,λ

)
P{X > i} =

∞∑
i=0

(
(i+ 1)r,λ − (i)r,λ

)(
1− P{X ≤ i}

)
=

∞∑
i=0

(
(i+ 1)r,λ − (i)r,λ

)(
1− FX(i)

)
.

Therefore, by (13), we obtain the following theorem.

Theorem 1. Let r be a positive integer, and let X be a discrete nonnegative integer-valued
random variable. Then the r-th degenerate moment of X is given by

E
[
(X)r,λ

]
=

∞∑
i=0

(
(i+ 1)r,λ − (i)r,λ

)(
1− FX(i)

)
. (14)



J. Kwon et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3847-3855 3850

Note that, when r = 1, we have

E[X] =
∞∑
i=0

(
1− FX(i)

)
=

∞∑
i=0

P{X > i}.

Assume that X and Y are discrete nonnegative integer-valued random variables with
their respective probability density functions pX(x) and pY (y).
Let T (x, y) = P{X > x, Y > y}, and let p(x, y) be the joint probability mass function of
X and Y . Now, we observe that

E[XY ] =
∞∑
x=0

∞∑
y=0

xyp(x, y) =
∞∑
x=1

∞∑
y=1

xyp(x, y) (15)

=
∞∑
x=1

∞∑
y=1

x∑
i=1

y∑
j=1

p(x, y) =
∞∑
i=1

∞∑
j=1

∞∑
x=i

∞∑
y=j

p(x, y)

=
∞∑
i=0

∞∑
j=0

∞∑
x=i+1

∞∑
y=j+1

p(x, y) =
∞∑
i=0

∞∑
j=0

P{X > i, Y > j}

=
∞∑
i=0

∞∑
j=0

T (i, j) =
∞∑
x=0

∞∑
y=0

T (x, y).

Therefore, by (15), we obtain the following theorem.

Theorem 2. Let X and Y be discrete nonnegative integer-valued random variables. Then
we have

E[XY ] =
∞∑
x=0

∞∑
y=0

T (x, y),

where T (x, y) = P{X > x, Y > y}.

If X and Y are independent, then we note that

E[XY ] = E[X]E[Y ] =

∞∑
x=0

∞∑
y=0

xypX(x)pY (y).

Note that

(i+ 1)r,λ − (i)r,λ =

r∑
j=0

(
r

j

)
(i)r−j,λ(1)j,λ − (i)r,λ (16)

=

r∑
j=1

(
r

j

)
(i)r−j,λ(1)j,λ, (r ≥ 1).

Thus, by (14) and (16), we get

E
[
(X)r,λ

]
=

∞∑
i=0

(
(i+ 1)r,λ − (i)r,λ

)(
1− FX(i)

)
(17)



J. Kwon et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3847-3855 3851

=

∞∑
i=0

r∑
j=1

(
r

j

)
(i)r−j,λ(1)j,λ

(
1− FX(i)

)
=

∞∑
i=0

r−1∑
j=0

(
r

j + 1

)
(i)r−1−j,λ(1)j+1,λ

(
1− FX(i)

)
.

Therefore, by (17), we obtain the following theorem.

Theorem 3. Let r be a positive integer, and let X be a discrete nonnegative integer-valued
random variable. Then the r-th degenerate moment of X is given by

E
[
(X)r,λ

]
=

∞∑
i=0

r−1∑
j=0

(
r

j + 1

)
(i)r−1−j,λ(1)j+1,λ

(
1− FX(i)

)
.

Assume that X,Y are discrete nonnegative integer-valued random variables. Let r1, r2
be positive integers. Then we have

E
[
Xr1Y r2

]
=

∞∑
x=0

∞∑
y=0

xr1yr2p(x, y) =

∞∑
x=1

∞∑
y=1

xr1yr2p(x, y) (18)

=

∞∑
x=0

∞∑
y=0

(x+ 1)r1(y + 1)r2p(x+ 1, y + 1)

=

∞∑
x=0

∞∑
y=0

x∑
i=0

(
(i+ 1)r1 − ir1

) y∑
j=0

(
(j + 1)r2 − jr2

)
p(x+ 1, y + 1)

=

∞∑
i=0

∞∑
j=0

(
(i+ 1)r1 − ir1

)(
(j + 1)r2 − jr2

) ∞∑
x=i

∞∑
y=j

p(x+ 1, y + 1)

=
∞∑
i=0

∞∑
j=0

(
(i+ 1)r1 − ir1

)(
(j + 1)r2 − jr2

)
P{X > i, Y > j}

=
∞∑
i=0

∞∑
j=0

(
(i+ 1)r1 − ir1

)(
(j + 1)r2 − jr2

)
T (i, j)

=

∞∑
x=0

∞∑
y=0

(
(x+ 1)r1 − xr1

)(
(y + 1)r2 − yr2

)
T (x, y).

Therefore, by (18), we obtain the following theorem.

Theorem 4. Let r1, r2 be positive integers, and let X,Y be discrete nonnegative integer-
valued random variables. Then we have

E
[
Xr1Y r2

]
=

∞∑
x=0

∞∑
y=0

(
(x+ 1)r1 − xr1

)(
(y + 1)r2 − yr2

)
T (x, y),

where T (x, y) = P{X > x, Y > y}.



J. Kwon et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3847-3855 3852

Assume that X1, X2, . . . , Xk are discrete nonnegative integer-valued random variables.
The joint probability mass function of X1, X2, . . . , Xk is defined by

p(x1, x2, . . . , xk) = P{X1 = x1, X2 = x2, . . . , Xk = xk}. (19)

The joint cumulative distribution function of X1, . . . , Xk is given by

FX1,X2,...,Xk
(a1, a2, . . . , ak) = P{X1 ≤ a1, X2 ≤ a2, . . . , Xk ≤ ak}. (20)

Let
T (x1, x2, . . . , xk) = P{X1 > x1, X2 > x2, . . . , Xk > xk}. (21)

For r1, r2, . . . , rk ∈ N, we have

E
[
Xr1

1 Xr2
2 · · ·Xrk

k

]
=

∞∑
x1=0

∞∑
x2=0

· · ·
∞∑

xk=0

xr11 xr22 · · ·xrkk p(x1, x2, . . . , xk) (22)

=
∞∑

x1=1

∞∑
x2=1

· · ·
∞∑

xk=1

xr11 xr22 · · ·xrkk p(x1, x2, . . . , xk)

=

∞∑
x1=0

∞∑
x2=0

· · ·
∞∑

xk=0

(x1 + 1)r1 · · · (xk + 1)rkp(x1 + 1, · · · , xk + 1)

=

∞∑
x1=0

∞∑
x2=0

· · ·
∞∑

xr=0

x1∑
i1=0

(
(i1 + 1)r1 − ir11

)
· · ·

xk∑
ik=0

(
(ik + 1)rk − irkk

)
p(x1 + 1, · · · , xk + 1)

=

∞∑
i1=0

∞∑
i2=0

· · ·
∞∑

ik=0

k∏
j=1

(
(ij + 1)rj − i

rj
j

) ∞∑
x1=i1

∞∑
x2=i2

· · ·
∞∑

xk=ik

p(x1 + 1, x2 + 1, · · · , xk + 1)

=

∞∑
i1=0

∞∑
i2=0

· · ·
∞∑

ik=0

k∏
j=1

(
(ij + 1)rj − i

rj
j

)
P{X1 > i1, X2 > i2, . . . , Xk > ik}

=
∞∑

i1=0

∞∑
i2=0

· · ·
∞∑

ik=0

k∏
j=1

(
(ij + 1)rj − i

rj
j

)
T
(
i1, i2, . . . , ik

)
=

∞∑
x1=0

∞∑
x2=0

· · ·
∞∑

xk=0

k∏
j=1

(
(xj + 1)rj − x

rj
j

)
T (x1, x2, . . . , xk).

Therefore, by (22), we obtain the following theorem.

Theorem 5. Let r1, r2, · · · , rk be positive integers, and let X1, X2, . . . , Xk be discrete non-
negative integer-valued random variables. Then the expectation of the monomial Xr1

1 Xr2
2 · · ·Xrk

k

is given by

E
[
Xr1

1 Xr2
2 · · ·Xrk

k

]
=

∞∑
x1=0

∞∑
x2=0

· · ·
∞∑

xk=0

k∏
j=1

(
(xj + 1)rj − x

rj
j

)
T (x1, x2, . . . , xk),

where T (x1, x2, . . . , xk) = P{X1 > x1, X2 > x2, . . . , Xk > xk}.



REFERENCES 3853

3. Conclusion

In recent years, many researchers have investigated various stuffs from probabilistic
perspectives and obtained quite a few interesting results (see [1–27] and the references
therein).

Let X be a discrete nonnegative integer-valued random variable. Then we showed that
the r-th degenerate moment of X is given by

E
[
(X)r,λ

]
=

∞∑
i=0

(
(i+ 1)r,λ − (i)r,λ

)(
1− FX(i)

)
=

∞∑
i=0

r−1∑
j=0

(
r

j + 1

)
(i)r−1−j,λ(1)j+1,λ

(
1− FX(i)

)
.

Let r1, r2, · · · , rk be positive integers, and let X1, X2, . . . , Xk be discrete nonnegative
integer-valued random variables. Then we proved that the expectation of the monomial
Xr1

1 Xr2
2 · · ·Xrk

k in X1, X2, . . . , Xk is given by

E
[
Xr1

1 Xr2
2 · · ·Xrk

k

]
=

∞∑
x1=0

∞∑
x2=0

· · ·
∞∑

xk=0

k∏
j=1

(
(xj + 1)rj − x

rj
j

)
T (x1, x2, . . . , xk),

where T (x1, x2, . . . , xk) = P{X1 > x1, X2 > x2, . . . , Xk > xk}.
Indeed, we derived this first for k = 2 and r1 = 1, r2 = 1, then for k = 2 with r1, r2 any

positive integers, and finally for the general case of any k and any r1, r2, · · · , rk. Certain
interesting applications of these results will be treated in a forthcoming paper.

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