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17

ON THE MOMENTS OF NORMAL PROBABILITY DISTRIBUTION

by Lucia Căbulea

Abstract: In this paper the author present a method for finding an explicit expression of the

( )

th

r

r

₁

,

₂ moments of the two-dimensional random vector having the normal distribution. Also is establish the recurrence relation (12) for the central moments of the order

( )

r1,r2 of the

random vector

(

Y₁,Y₂

)

.

1.Consider a two-dimensional random vector Y =

(

Y₁,Y₂

)

, and let be

(

y1, y2;x1,x2

)

F , the probability distribution of

(

Y₁,Y₂

)

, where

(

y₁,y₂

)

is any point of the Euclidian space R2 and

(

x₁,x₂

)

is a real two-dimensional parameter, varying in a parameter space Ω₂, which is a subset of R2.

Let f = f

(

y₁,y₂

)

be a real-valued function defined and bounded on

R

2 such that the mean value of the random variable f

(

Y₁,Y₂

)

exists.

As is well known, this mean value can be expressed by the improper Stieltjes integral of

(

y₁,y₂

)

with respect to F

(

y₁,y₂;x₁,x₂

)

.

(1)

[

(

)

]

=

∫

(

) (

)

2

2 1 2 1 2 1 2

1, , , ; ,

R

x x y y dF y y f Y

Y f E

Assuming that the random vector

(

Y₁,Y₂

)

is of continuous type, having probability densities ρ

(

y₁,y₂;x₁,x₂

)

, then we have

(

1 2 1 2

)

(

1 2 1 2

)

1 2

1 2

, ; , ,

;

,y x x u u x x du du

y F

y y

∫ ∫

_∞

− ∞−

= ρ

and the mean value by (1) is

(2)

[

(

_{1 2}

)

]

(

_{1 2}

) (

_{1 2 1 2}

)

_{1 2}

2

, ; , ,

,Y f y y y y x x dydy

Y f E

R

∫

= ρ

Considering that

(

) (

) (

1

)

2

2 2 1 1 2 1,

r r

a y a y y y

18

where r₁,r₂ ∈N and

(

y₁,y₂

)

∈R2, we have the moment of order

( )

r₁,r₂ , of the point

(

a₁,a₂

)

define if it exists- such the mean value of vectors

(

) (

) (

1

)

2

2 2 1 1 2 1,

r r

a Y a Y Y Y

f = − −

(3)

(

) (

[

) (

1

)

2

]

2

1, 1, 2 1 1 2 2

r r

r

r a a =E Y −a Y −a

ν .

For

(

a₁,a₂

) ( )

= 0,0 , we have the ordinary moment of order

( )

r₁,r₂ of the random vector

(

Y₁,Y₂

)

.

(4)

( )

[

1 2

]

2 1 2

1, 0,0 , 1 2

r r r

r r

r =ν = EY Y

ν

If ν_1,0 =E

( )

Y₁ and ν_0,1=E

( )

Y₂ we can write expression for the central moment of the order

( )

r₁,r₂ of the random vector

(

Y₁,Y₂

)

.

(5)

[

(

) (

1

)

2

]

2

1, 1 1,0 2 0,1

r r

r

r E Y ν Y ν

µ = − − .

2. Let us assume that random variables Y₁,Y₂ are not independent and the two-dimensional random vector Y having the normal distribution with the probability density

(6)

(

)

(

)

(

)(

)

_

   

    

  

  

 − + − −

−   

 − −

− ⋅

⋅ − =

2

2 2 2

2 1

2 2 1 1 2

1 1 1 2

2 2

1 2

1 2 1

2 1

2 1 exp

1 2

1 ,

; ,

σ

σ

σ

ρ

σ

ρ

ρ

σ

πσ

ρ

x y x

y x y x

y x

x y y

where x₁,x₂ ∈R, σ₁ >0,σ₂ >0,ρ∈

[ ]

−1,1 , and

(

y₁,y₂

)

∈R2. It is easy to see that ρ

(

y₁,y₂;x₁,x₂

)

>0,

(

y₁,y₂

)

∈R2 and

(

₁, ₂; ₁, ₂

)

_{1 2} 1

2

=

∫

y y x x dydy

R

ρ , if we change the variables

(7)   

+ == 2+ 2 2

1 1 1

x u y

x u y

σσ

19

(

)

( )

0

, , 2 1 2 2 1 1 2

1 _{= >}

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ = σ σ v y u y v y u y v u D y y D

For r₁ =r₂ =1, the central moment of the random vector

(

Y₁,Y₂

)

become

(

)(

)

[

1 1,0 2 0,1

]

(

1 1

)(

2 2

) (

1 2 1 2

)

1 2 1

, 1

2

1 E Y Y y x y x y ,y ;x,x dydy

R

∫

− − = − − = ν ν ρ µ

By using the formula (6) we get

(

)(

)

(

)

(

)(

)

2 1 2 2 2 2 2 1 2 2 1 1 2 1 1 1 2 2 2 1 1 2 2 1 1 , 1 2 1 2 1 exp 1 2 1 2 dy dy x y x y x y x y x y x y R             − + − − − ∫         −     − − − − − ⋅ − = σ σ σ ρ σ ρ ρ σ πσ µ

According to the relation (7) we have

( )

( )

∫

∫

_        − = − − − − R R v u v dv du ue ve 2 2 2 1 2 2 2 2 1 1 , 1 1 2 ρ ρ ρ π σ σ µ

For u−ρv=t 1−ρ2 we get (8) µ_1,1 = ρσ₁σ₂

We observe that the parameter ρ is the corelation coefficient of random variables Y₁

and Y₂.

For the ordinary moment of order (1,1) we have

(

)

(

)(

)

∫                     − + − − −     − − − ⋅ ⋅ − =

2 1 2

2 2 2 2 2 1 2 2 1 1 2 1 1 1 2 2 1 2 2 1 1 , 1 2 1 2 1 exp 1 2 1 R dy dy x y x y x y x y y y σ σ σ ρ σ ρ ρ σ πσ ν

and using relation (7) results

(

u x

)(

v x

)

v

(

u

(

v

)

)

dudv

R         − − − − + + −

=

∫

2

20

If u−ρv =t 1−ρ2 we get

(9) ν_1,1 = ρσ₁σ₂ +x₁x₂.

3. We should remark that in the special case ρ =0 results µ_1,1 =0 and the random variables Y₁ and Y₂ are independent.

In this case (ρ =0) for r₁ =r₂ =1, the ordinary moment of the random vector

(

Y₁,Y₂

)

become

[ ]

∫

(

) (

)

    

 

₋ − ₋ −

= =

2

2 1 2 2

2 2 2 2

1 2 1 1 2

1 2 1 2

1 1 , 1

2 2

exp 2

1

R

dy dy x y x y y

y Y

Y E

σ σ

σ πσ ν

and using relation (7) we have

(10)

(

)(

)

2 2 _{1 2}

2 2 1 1 1

, 1

2 2

2

2 1

x x dudv e

x v x u

v u

R

= +

+

=

∫

σ σ − −

π

ν .

Analogous we have

(

) (

)

(

) (

)

∫

_

  

 

₋ − ₋ −

− −

=

2

2 1 2 2

2 2 2 2

1 2 1 1 2

2 2 2 1 1 2 1 2

, 2

2 2

exp 2

1

R

dy dy x y x y x

y x y

σ σ

σ πσ µ

and using the relation (7) we get

(11)

(

)

2

2 1 2 2 2 1 2 ,

2 σ σ σ σ

µ = =

It should be noticed that for r₁ or r₂ being a non-negative impare integer we have 0

1 2 , 1 2k+ k+ =

µ where k₁∈N*,k₂∈N* we get recurrence formula for the central moments

(12) _,

(

₁

)(

₂

)

₁2 ₂2 _{2, 2}

2 1 2

1 r = −1 −1 r − r −

r r r σ σ µ

µ

For the special case r₁ = r₂ =2k,k∈N* we have

(13) µ_2k,2k =

(

2k−1

) (

2 2k−3

)

2⋅...⋅32⋅12⋅

(

σ₁σ₂

)

2k,k∈N*

21

(14)

( )

(

(

)

)

(

)

_

 

  

  

 − ₋ −

− − −

=

= =

2

2 2 2

1 1 1 2 2

1

2 2

2 1 2 1 2

1

1 2

1 exp

1 2

1

; , ; ,

σ ρ σ

ρ ρ

σ π

ρ ρ ρ

x y x y x

y x x y y y

y

and

(15)

( )

(

(

)

)

(

)

_

 

  

  

 − ₋ −

− − −

=

= =

2

1 1 1

2 2 2 2 2

2

1 1

2 1 2 1 1

2

1 2

1 exp

1 2

1

; , ; ,

σ ρ σ

ρ ρ

σ π

ρ ρ ρ

x y x y x

y x x y y y

y

where ρ

(

y₁;x₁

)

is the probability density of random variable Y₁

(16)

(

)

(

)

    

 

 ₋

−

= ₂

1 2 1 1

1 1

1

2 exp 2

1 ;

σ σ

π

ρ y x y x

and

ρ

(

y₂;x₂

)

is the probability density of random variable Y₂

(17)

(

)

(

)

    

 

 ₋

−

= ₂

2 2 2 2

2 2

2

2 exp 2

1 ;

σ σ

π

ρ y x y x

We find that

(18)

( )

( )

(

_{2 2}

)

2 1 1 1 2 1 1 2

1 y y y y dy x y x

Y E

R

− +

= =

∫

σσ ρ ρ

and

(19)

( )

( )

(

_{1 1}

)

1 2 2

2 1 2 2 1

2 y y y y dy x y x

Y E

R

− +

= =

∫

σ σ ρ ρ

It should be noticed that the regresion between the random variables Y₁ and Y₂ for the two-dimensional random vector Y =

(

Y₁,Y₂

)

it is a liniary function.

REFERENCES

22

[3] Feller W., An Introduction to Probability Theory and Its Applications, Wiley, New York, 1968.

[4] Gnedenko B.V. , The Theory of Probability, Mir Publisher, Moskow, 1968. [5] Iosifescu M., Mihoc Gh., Theodorescu R., Teoria probabilităţilor şi staistică matematică, Ed. Tehnică, Bucureşti, 1966.

[6] Onicescu O., Probabilităţi şi procese aleatoare, Ed. Ştiinţificăşi Enciclopedică, Bucureşti, 1977.

[7] Purcaru I., Matematici generale şi elemente de optimizare, Ed. Economică, Bucureşti, 1997.

[8] Stancu D.D., Recurrence relations for the central moments of some discrete probability laws, Studia Univ. Babeş-Bolyai, Ser. Math.-Mech., 15,1970,55-62. [9] Stancu D.D., A method for computing the moments, Ser. Math.-Phys., 1, 1967, 45-54.

[10] Ventsel H., Théorie des probabilités, Ed. Mir Moscow, 1973.

Author:

Lucia Căbulea, „ 1 Decembrie 1918” University of Alba Iulia, Romania,