USED OF INTERPOLATORY LINEAR POSITIVE OPERATORS FOR
CALCULUS THE MOMENTS OF THE RELATED PROBABILITY
DISTRIBUTIONS
by
Lucia A. C
ă
bulea and Mihaela Todea
Abstract. In this paper using probabilistic methods for constructing linear positive operators and the Newton interpolation formula on representing a linear interpolatory positive operators by means of the factorial moments of the related probability distribution and the finite differences. By means of such representations we deduce explicit formulas for the ordinary moments of the corresponding discrete probability distributions
.
Keywords: probabilistic methods, interpolatory linear positive operators, moments of
the related probability distributions, factorial moments, ordinary moments.
1.Consider a sequence of two-dimensional random vectors
Yn={(Yn1, Yn2)} and let Fn (y1, y2; x1, x2) be the probability distribution of (Yn1, Yn2),
where (y1, y2) is any point of the Euclidean space R2 and (x1, x2) is a real
two-dimensional parameter varying in a parameter space Ω2, which is a subset of R2.
We suppose that (x1, x2) represents the mean value of this distribution i.e.
x1 =
∫
2
) , ; ,
( 1 2 1 2
1
R
n y y x x
dF y
x2 =
∫
2
) , ; ,
( 1 2 2
2
R
x n y y x x
dF y
If f = f(y1,y2) is a real-valued function defined and bounded on R2 such that the
mean value of the random variable f(Yn1,Yn2) exists for n = 1,2,…, there fore this
mean value can be expressed by the improper Stieltjes integral of (y1,y2) with respect
to Fn (y1,y2; x1,x2).
(1) E [f (Yn1, Yn2)] = Pn (f ; x1,x2) =
∫
2
) , ; , ( ) ,
( 1 2 1 2 1 2
R
n y y x x
If we suppose that the radom vector (Yn1,Yn2) is of discrete type, one may
observe that its distribution function :
Fn (y1, y2; x1, x2) = P [Yn1 ≤ y1, Yn2 ≤ y2; x1, x2]
is a step function so that P[Yn1=y1, Yn2=y2; x1, x2] is zero at every point of R2 except at
a finite or a countable infinite such point (jump point) is taken with a positive probability (jump):
pn (ak1, ak2) = pn (ak1, ak2; x1, x2) = P [Yn1 = ak1, Yn2 = ak2; x1,x2]
satisfying the condition
∑
p (a 1,a 2;x1,x2)=1k
k k
n .
The corresponding distribution function is: Fn (y1, y2; x1, x2) =
∑
) (
2 1 2 1, ; , )
(
k
k k
n a a x x
p
where the summation is extended now over all points (ak1, ak2) such that
ak1 ≤ y1, ak2 ≤ y2.
Consequently, in this discrete case we are able to write down the following expression for the operator (1)
(2) Pn ( f ; x1, x2 ) =
∑
) (2 1 2 1 2
1, ) ( , ; , )
(
k
k k n k
k a p a a x x
a f
It is easy to see that the operator Pn (f; x1, x2) defined by (1), or in particular by
(2) is a positive linear operator.
2.We shall now make use of an important method for constructing concrete
operators of this useful for computing the moments of the related probability distributions.
Consider a sequence of two-dimensional random vectors {(Xk1, Xk2) = Xk}
and let us assume that the components Yn1, Yn2 of the random vector Yn represent the
arithmetic means of the first n components Xk1, Xk2
(k = 1,2,…,n) that is:
(3) Ynr =
n
1
[X1r +X2r +…+ Xnr] (r = 1,2).
i) Let us suppose first that the components Yn1 , Yn2 have the binomial
distribution. Now referring to (2) we obtain the operator of Bernstein
(4)
∑∑
= −
=
−
−
−
−
− = n
k k n
k
k k n k
k n
n k n k f x
x x x k
k n k
n x
x f B
0 0
2 1 2
1 2 1 2
1
1 2
1
1 1
2
2 1 2
1 (1 ) ,
) , ; (
ii) If we consider that X1i, X2i, …, Xni (i=1,2) has a Poisson distribution with
the parameters xi (i=1,2) therefore Yni have a Poisson distribution with the parameters
nxi (i=1,2) and we obtain the operator
(5)
=
∑
∞= + −
n k n k f k
k nx nx e
x x f P
k k x
k k
x x n n
2 1
2 1
2
0 ,
) ( 2
1 ,
! !
) ( ) ( )
, ; (
2 1
2 1
which represents an extension to 2 variables of an operator studied early by Favard [3] and Szasz [15].
iii) If we presuppose that X1i, X2i, …, Xni (i=1,2) has a geometric distribution
then Yni have a Pascal distribution and we obtain the operator
(6) − − + − + − =
∑
∞ = n k n k f x x x x k k n k k n x x fP n k k
k k
n 1 2 1 2 1 2
2 2 0 , 1 1 2
1 ( ) (1 ) (1 ) ,
1 1
) , ;
( 1 2
2 1
iv) It should be observed that if we replace xi by
i
x
+
1
1
, (i=1,2) in formula
(6), then we arrive the operator (7)
+ + + − + − = ∞ + + =
∑
n k n k f x x x x k k n k k n x x f P k n k n k k k k n 2 1 2 1 2 1 0 , 2 2 1 1 2 1 , ) 1 ( ) 1 ( 1 1 ) , ; ( 2 1 2 1 2 1which has been considered first by Baskakov [1].
v) If the random variables X1i, X2i, …, Xni (i=1,2) are not independent and
identically distributed we obtain the operator of Stancu
(8)
∑
= + = n k k x k k n n n k n k f x x W x x f P 0 2 1 2 , 2 1 ] [ 2 1 2
1 ( , ; ) ,
) , ; ( α α where
∏
=− +∏
−= + −∏
−= − − − + = 1 0 1 0 1 0 2 1 2 2 1 1 , 2 , 1 1 2 2 2 1 2 1 21 ( ) ( ) (1 )
) ( ) ; , (
k k n k k
k k n x k k
n x x x x
n B C x x W ν ν α ν ν α µ µα α ) ) 1 ( 1 )...( 2 1 )( 1 ( )
(n = +α + α + n− α
B ,
)! ( ! ! ! 2 1 2 1 ,2 1 k k n k k n Cnk k
− − =
3.Now we consider of an interpolation polynomial of Newton-Bierman type
for two variables [8]
( )
0,0 ! ! ) ( ) ( ) , ;( 1 2
2 1 2 1 , 1 , 1
0 1 2
] [ 2 ] [ 1 2 1 f i i nt nt t t f
N i i
n n n i i i i ∆ =
∑
= +where )(ntk)[ik] =ntk(ntk −1)...(ntk −ik +1 (k=1,2) while
( )
− − − = ∆∑∑
= = + n i n i f i i f i i i i n n 2 2 1 1 2 2 0 1 1 0 , 1 ,1 0,0 ( 1) ,
1 1 2 2 2 1 2
1 ν ν
ν ν
ν ν
ν
ν represents the finite
partial difference of order (i1, i2) of the function f, with the steps n
1
and the starting point (0,0).
(9)
( )
0,0 !! ,
; 1 2
2 1 2 1 , 1 , 1 0 1 2
] [ 2 ] [ 1 2 1 f i i y y n y n y f
N i i
n n n i i i i ∆ =
∑
= +This polynomial satisfies the interpolating properties
= n k n k f n k n k f
N ; 1, 2 1, 2 for k1 = 0, 1, …, n , k2 = 0, 1, …, n-k1.
By using the formula (9) we can find for the mean value of the random variable n y n y f
N ; 1, 2 where (y
1, y2) has the probability distribution function
Fn(y1, y2; x1, x2), the following representation
(10)
( )
0,0! ! ) , ; , ( ,
; 1 2
2 1 2 1 2 , 1 , 1 0 1 2
] , [ 2 1 2 1 2 1 f i i m x x y y dF n y n y f
N i i
n n n i i i i n R ∆ =
∑
∫
+ =in terms of the factorial moments
∫
= 2 2 1 21 ( 1, 2; 1, 2)
] [ 2 ] [ 1 ] , [ R n i i i
i y y dF y y x x
m
If the random vector Yn is of discrete type then
(11)
∑
∑
( )
= + = + = ∆ n k k i i n n i i n k k k k n f i i m n k n k f p 0 , 1 , 1 2 1 ] , [ 0 2 1 , ; 2 1 2 1 2 1 2 1 2
1 , ! ! 0,0
where
∑
= + = n k k k k n i i i
i k k p
m 0 , ; ] [ 2 ] [ 1 ] , [ 2 1 2 1 2 1 2 1 .
4. i) With the aid of formula (11) we can give the following representation of
the operators of Bernstein type (4) in terms of finite differences
(12)
( )
0,0! ! )
, ;
( 1 2 1 2
2 1 2 1 , 1 , 1 2 1 0 1 2
] [
2
1 x x f
i i n x x f
B i i
n n i n i n i i i i
n =
∑
∆= +
+
which enables us to find the factorial moments
(13) 1 2 1 2
2
1 1 2
] [ ] , [ i i i i i
i n x x
m = +
ii) The operator Pn(f;x1, x2) defined by (5) in terms of finite differences have
the following representation
(14)
∑
( )
= ∆ = 0 , , 1 , 1 2 1 2 1 2 1 2 1 2 1 2 1 0 , 0 ! ! ) , ; ( i i i i n n i i n f i i z z z z f P
if we assume that xi depends on n such a way for n →∞ we have nxi→ zi >0 , (i=1,2)
and we obtain the factorial moments
(15) 1 2
2
1, ] 1 2
[
i i i
i z z
iii) The operators of Stancu defined by (8) in terms of finite differences have the following representation (16)
( )
∑
=∏
+ = + + − ∆
= n
i i
i i
n n i
i n
n x x x i f
i i B
C x
x f P
0
2
1
, 1 , 1 2
1 ,
2 1 ] [
2 1
2 1 2
1
0 , 0 )
) 1 ( )...( (
) , ( )
, ; (
ν ν ν ν ν
α α α
and we obtain the factorial moments
(17) ( )...( ( 1) )
) (
2
1 2 1
] [
] , [
2 1 2
1 ν α ν ν α
ν ν + + −
+
=
∏
= +
i x x
x i i B
n m
i i
i i
5. For the function
(
)
1 2 1 22 1 2
1,
r r r r y y
n y y
f = + we have 1,2
( )
0,0 101 2021 , 1
r i r i i
i
n n
f =∆ ∆
∆ .
In this case the operator defined by (12) permits us to find the express of the ordinary moments
) , ; ( )
, ;
( 1 2 , 1 2
2
1 n x x
m x x f
Bn = r r
that is
(18)
( )( )
1 1 2 2 1 21
1 2
2 2
1 1 2
2 1
0 0 1 2
1
, ( ; , ) 0 0
i i r i r i r
i r
i r
r x x
i i n i n x
x n
m ∆ ∆
−
=
∑∑
= =
if ∆m0r = 0 for m > r and =0
ν
m
for ν > m.
Analogous the operator defined by (14) enables us to find
(19)
( )( )
2 21
1 2
2
1 1 2 1 2
1 ( ; , ) ! ! 0 0
0 0 1 2 2 1 2
1 ,
r i r
i r
i
r i i i
r r
i i
z z z
z n
m =
∑∑
∆ ∆= =
REFERENCES
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249-251.
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computing the moments of the related probability distributions, Studia Univ. “Babeş
-Bolyai”, Math.,4(1994), 79-91.
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Authors:
Lucia A. Căbulea, “1 Decembrie 1918” University of Alba Iulia, Romania,
Departament of Mathematics, e-mail: lcabulea@uab.ro.
Mihaela Todea, “1 Decembrie 1918” University of Alba Iulia, Romania, Departament
