Further results and applications on continuous random variables

(1)

https://doi.org/10.26637/MJM0703/0011

Further results and applications on continuous

random variables

Mohamed DOUBBI BOUNOUA

1

, Zoubir DAHMANI

2

* and Zakaria BEKKOUCHE

3

Abstract

New results and new applications of fractional calculus for continuous random variables are presented. New expectation and variance identities of orderα≥1are established. Under a new fractional normalisation technique, other weighted random variable inequalities are generated and some classical results are deduced as special cases.

Keywords

Integral inequalities, Riemann-Liouville integral, random variable, fractional expectation, fractional variance, fractional moment.

AMS Subject Classification 26D15, 26A33, 60E15.

1_{Department of Mathematics, Shanghai China, China.}

2_{Laboratory LPAM, UMAB University, Algeria.}

3_{School of Mathematics and Statistics, Central China Normal University, China.}

*Corresponding author:1_{doubbibounoua.mohamed@yahoo.fr;} 2_{zzdahmani@yahoo.fr;}3_{bezack@mails.ccnu.edu.cn}

Article History: Received24March2019; Accepted09May2019 c2019 MJM.

1. Introduction

The integral inequalities are very important in physics and applied sciences. For some applications of this theory, we refer the reader [2,4,5,8–11]. Let us now introduce some papers that have motivate the present work. We begin by [1] where, T. Cacoullos and V. Papathanasiou established the following nice covariance identity

Cov(h(X),g(X)) =E(z(X)g0(X)), (1.1) whereX is a continuous random variable ( CRV, for short) with support an interval(a,b),−∞≤a<b≤∞, f is

proba-bility density function ( pdf, for short) ofX,gandhare two continuous functions withσ2=V(X),|E(z(X)g0(X))|<∞

and

z(x):= 1

f(x)

x

Z

a

(E[h(X)]−h(t))f(t)dt. (1.2)

Under some other conditions, the same authors [2] proved that

Var[g(X)]≥E

2_[_z₍_X₎_g0₍_X_)]

E[z(X)h(X)] , (1.3)

They also established the following theorem [3]:

Theorem 1.1. For every absolutely continuous function h with h0>0,the inequality:

Var[g(X)]≤E[z(X)

h0(X)(g

0₍

X))2] (1.4)

holds, with equality iff g=c1h+c2, where

z f =

x

Z

a

(E(h)−h(t))f(t)dt. (1.5)

Very recently, based on the notions of [4], Z. Dahmani [5] proved the following theorem:

Theorem 1.2. Let X be a random variable with a p.d.f.

defined on[a,b],such thatµ=E(X),σ2=Var(X)and w∈ C([a,b]);Rx

a(b−t)α−1(µ−t)f(t)dt= (b−x)α−1σ2w(x)f(x).

Then, we have

Var_g(X),α≥ σ4(X) VarX,αE

2

(2)

Motivated by the papers in [2,3,5,8], in this work, we fo-cus our attention on the application of the Riemann-Liouville fractional integration on random variables. We use some re-cently introduced concepts on continuous random variables to establish new identities on continuous random variables. We prove new lower bounds for the variances as well as for the expectations. For our results, some classical and excellent results of [1–3], that correspond to the standard integration of orderα=1,are generalised for anyα≥1.

2. Preliminaries

Definition 2.1. [7]The Riemann-Liouville fractional integral operator of orderα>0,for a continuous function f on[a,b]

is defined as

Jα

a[f(t)] =

1

Γ(α)

t

Z

a

(t−τ)α−1f(τ)dτ,α>0,a<t≤b. (2.1)

For anyα >0 and any positive continuous functionw defined on[a,b],we recall the definitions [4]:

Definition 2.2. The fractional w−weighted expectation func-tion of orderα,for a CRV X with a pdf f defined on[a,b]is given a by:

EX,α,w(t) =

1

Γ(α)

t

Z

a

(t−τ)α−1τw(τ)f(τ)dτ,a≤t<b,

(2.2)

Definition 2.3. The fractional w−weighted expectation func-tion of orderα,for X−E(X)is given by

EX−E(X),α,w(t) =

1

Γ(α)

t

Z

a

(t−τ)α−1(τ−E(X))

×w(τ)f(τ)dτ. (2.3)

Definition 2.4. The fractional w−weighted variance function of orderα,for X is defined as

σ_X2_,_α_,_w(t) = 1 Γ(α)

t

Z

a

(t−τ)α−1(τ−E(X))2w(τ)f(τ)dτ.

(2.4)

Definition 2.5. The fractional w−weighted expectation of orderα for X is defined as

EX,α,w=

1

Γ(α)

b

Z

a

(b−τ)α−1τw(τ)f(τ)dτ. (2.5)

Definition 2.6. The fractional w−weighted variance of order αfor X is given by:

σ_X2_,_α_,_w= 1

Γ(α)

b

Z

a

(b−τ)α−1(τ−E(X))2w(τ)f(τ)dτ.(2.6)

3. Main Results

We begin this section by proving the following theorem. We have:

Theorem 3.1. Let X be a CRV with support an interval

[a,b],−∞<a<b<∞.Suppose that X has a pdf f and

let w be a positive continuous function on[a,b].Then, the following equality holds for anyα≥1 :

Ezg0_,_α_,_ω=E_gh_,_α_,_w−E_h_,_α_,_wE_g_,_α_,_w, (3.1)

where g is an absolutely continuous function withE_zg0_, α,ω

<

∞,J_aαw f(b) =1, h is a given function and z satisfies:

z(t) = 1

(b−t)α−1_w₍_t₎_f₍_t₎

× t

Z

a

(b−u)α−1w(u)f(u)(E_h,α,w−h(u))du. (3.2)

Proof.

We have

E_zg0_, α,ω=

1

Γ(α)

b

Z

a

(b−t)α−1g0(t) 1

(b−t)α−1_w₍_t₎_f₍_t₎

× t

Z

a

(b−u)α−1w(u)f(u)(Eh,α,w−h(u))du

×w(t)f(t)dt

= 1 Γ(α)

b

Z

a

g0(t)

t

Z

a

(b−u)α−1w(u)f(u)

(3)

Therefore,

E_zg0_, α,ω=

1

Γ(α)

g(t)

t

Z

a

×(Eh,α,w−h(u))du|tt==ba

− 1

Γ(α) Zb

a

g(t)

×(b−t)α−1w(t)f(t)(Eh,α,w−h(t))dt

= 1 Γ(α)

g(b)

b

Z

a

×(Eh,α,w−h(u))du

− 1

Γ(α) Zb

a

g(t)

×(b−t)α−1w(t)f(t)(Eh,α,w−h(t))dt

=g(b)

Eh,α,w

Γ(α)

b

Z

a

(b−u)α−1w(u)f(u)du

− 1

Γ(α)

b

Z

a

(b−u)α−1w(u)f(u)h(u)du

−

_E

h,α,w

Γ(α)

b

Z

a

g(t)(b−t)α−1w(t)f(t)dt

− 1

Γ(α)

b

Z

a

g(t)(b−t)α−1w(t)f(t)h(t)dt

=g(b)

Eh,α,wJaαw f(b)−Eh,α,w

−Eh,α,wEg,α,w+Egh,α,w.

Thanks to the hypothesisJα

aw f(b) =1, we obtain

Ezg0_,_α_,_ω=E_gh_,_α_,_w−E_h_,_α_,_wE_g_,_α_,_w. The proof is thus achieved.

Remark 3.2. If we takeα=1,w(x) =1in(3.1), we obtain

(1.1).

Lemma 3.3. Let X be a CRV with support an interval[a,b]

,−∞≤a<b≤∞,and a pdf f and suppose that w:[a,b]→

R+is a continuous function. Then, for anyα≥1,we have σ_g2,α,w=Ezg0,α,ω, (3.3) where g is an absolutely continuous function such that

E_zg0_,_α_,_ω<∞,g(b) =E(g)and z(t) = 1

(b−t)α−1_w₍_t₎_f₍_t₎

× t

Z

a

(b−u)α−1(E(g)−g(u))w(u)f(u)du.

Proof:

We have

E_zg0_,_α_,_ω= 1

Γ(α)

b

Z

a

(b−t)α−1g0(t) 1

(b−t)α−1_w₍_t₎_f₍_t₎

× t

Z

a

(b−u)α−1(E(g)−g(u))

×w(u)f(u)duw(t)f(t)dt

= 1 Γ(α)

b

Z

a

g0(t)

t

Z

a

(b−u)α−1(E(g)−g(u))

×w(u)f(u)dudt.

Hence, we get

Ezg0_,_α_,_ω= 1

Γ(α)g(t)

t

Z

a

(b−u)α−1(E(g)−g(u))

×w(u)f(u)du|b a−

1

Γ(α)

b

Z

a

g(t)

×(b−t)α−1(E(g)−g(t))w(t)f(t)dt

= 1 Γ(α)g(b)

b

Z

a

(b−u)α−1(E(g)−g(u))

×w(u)f(u)du− 1

Γ(α)

b

Z

a

(b−t)α−1

×(g(t)(E(g)−g2(t))w(t)f(t)dt.

We use the fact thatg(b) =E(g),then it yields that

Ezg0_,_α_,_ω= 1

Γ(α)

b

Z

a

(b−u)α−1(E2(g)−E(g)g(u))

×w(u)f(u)du− 1

Γ(α)

b

Z

a

(b−t)α−1

×(g(t)E(g)−g2(t))w(t)f(t)dt

= 1 Γ(α)

b

Z

a

(b−t)α−1((g2(t) +E2(g)

−2g(t)E(g))w(t)f(t)dt

= 1 Γ(α)

b

Z

a

(b−t)α−1(g(t)−E(g))2w(t)f(t)dt

=σ_g2,α,w.

Remark 3.4. If we take g(x) =x in Lemma 3.3, we obtain E(X) =b,and

(4)

with,

z(t) = 1

(b−t)α−1_w₍_t₎_f₍_t₎ t

Z

a

(b−u)α−1(E[X]−u)

×w(u)f(u)du

= 1

(b−t)α−1_w₍_t₎_f₍_t₎ t

Z

a

(b−u)α_w₍_u₎_f₍_u₎_du_. (3.5)

Now, taking three continuous functionsU,V,Qdefined on

[a,b],we prove the following result:

Theorem 3.5. Let U,V and Q be three continuous functions on[a,b] and Jα

aQ(b) =1,then

Jα

aUV Q(b)−JaαU Q(b)JaαV Q(b) =Jaα[(U−JaαU Q(b)) ×(V−Jα

aV Q(b))Q(b)

(3.6)

is valid, for anyα≥1.

Proof

We have

Jα

a[(U−JaαU Q(b)) (V−JaαV Q(b))] (b)

= 1

Γ(α)

b

Z

a

(b−t)α−1(U(t)−Jα

aU Q(b))

×(V(t)−Jα

aV Q(b))Q(t)dt

= 1

Γ(α)

b

Z

a

(b−t)α−1U(t)V(t)Q(t)dt−J

α

aV Q(b)

Γ(α)

× b

Z

a

(b−t)α−1U(t)Q(t)dt−J

α

aU Q(b)

Γ(α)

× b

Z

a

(b−t)α−1V(t)Q(t)dt+J

α

aU Q(b)JaαV Q(b)

Γ(α)

× 1

Γ(α)

b

Z

a

(b−t)α−1Q(t)dt

= Jα

aUV Q(b)−2JaαU Q(b)JaαV Q(b)

+Jα

aU Q(b)JaαV Q(b)JaαQ(b).

SinceJα

aQ(b) =1,then we ca write

Jα

a [(U−JaαU Q(b)) (V−JaαV Q(b))Q] (b)

= Jα

aUV Q(b)−JaαU Q(b)JaαV Q(b).

Taking into account the above theorem, we prove the following theorem:

Theorem 3.6. Let X be a continuous random variable with support an interval[a,b],−∞<a<b<∞,having a pdf f

and let w:[a,b]→_R+_{be a continuous function. Then, for} anyα≥1,we have

E_zg20_, α,w

E_zh0_, α,w

≤E_[_g₋_E

g,α,w]2,α,w,0<α, (3.7)

where g is an absolutely continuous function with Ezg0_,_α_,_ω<

∞and Jα

aw f(b) =1,h is a given function and z is given by

z(t) = 1

(b−t)α−1_w₍_t₎_f₍_t₎ t

Z

a

×(Eh,α,w−h(u))du. Proof

Thanks to (3.1), we observe that

E_zg0_,

α,ω = Egh,α,w−Eh,α,wEg,α,w

= Jα

aghw f(b)−Jaαhw f(b)Jaαgw f(b).

In (3.6), we takeU=g,V =h andQ=w f. Then, we observe that

Ezg0_,_α_,_ω = J_aα[(g−J_aαgw f(b)) (h−J_aαhw f(b))w f] (b)

= 1

Γ(α)

b

Z

a

(b−t)α−1(g(t)−Jα

agw f(b))

×(h(t)−Jα

ahw f(b))w f(t)dt.

Thanks to Cauchy-Shwarz inequality, we obtain

E_zg20_, α,ω

≤



 1

Γ(α)

b

Z

a

(b−t)α−1(g(t)−Jα

agw f(b)) 2

w f(t)dt 



×



 1

Γ(α)

b

Z

a

(b−t)α−1(h(t)−Jα

ahw f(b))w f(t)dt





= Jα

a

h

(g−Jα

agw f(b))

2_{w f}i₍_b₎_Jα

a

h

(h−Jα

ahw f(b))

2_{w f}i₍_b₎

= E₍_g₋_Jα

agw f(b))2,α,wE(h−Jα

ahw f(b))2,α,w

= E

(g−Eg,α,w)

2

,α,wE(h−Eh,α,w)

2

,α,w. (3.8)

Again, takingg=hin (3.1), we get

E_zh0_,

α,ω = Eh2_,_α_,_w− Eh,α,w 2

= E_h2_,_α_,_w−2 Eh,α,w 2

+ Eh,α,w 2

.

Thanks to the hypothesisJα

aw f(b) =1, it yields that

Ezh0_,_α_,_ω = E_h2_,_α_,_w−2 E_h,α,w

2

+ Eh,α,w

2 Jα

aw f(b)

= Jα

a h2w f

(b)−2Eh,α,wJaα(hw f) (b)

+ Eh,α,w 2

Jα

aw f(b)

= Jα

a

h

h2−2Eh,α,wh+ Eh,α,w

2i w f(b)

= Jα

a

h−Eh,α,w

2 w f

(b)

= E

(h−Eh,α,w)

2

(5)

Using (3.8) and (3.9), we deduce that

E_zg20_,_α_,_w

E_zh0_, α,w

≤E_[_g₋_E

g,α,w]2,α,w.

The proof is thus achieved.

Remark 3.7. If we take α=1and w(x) =1in(3.7),we obtain(1.3).

Lemma 3.8. Let X be a CRV with support an interval[a,b]

,−∞<a<b<∞,having a pdf f and let w:[a,b]→R+be

a continuous function. Then, for anyα≥1,we have E₍_g₋_E

g,α,w)2,α,w (3.10)

= 1

[Γ(α)]2 Z b

a

Z y a

(b−x)α−1(b−y)α−1(g(x)−g(y))2

×f(x)f(y)w(x)w(y)dxdy,

where g is an absolutely continuous function and Jα

aw f(b) =1. Proof.

We have

1 [Γ(α)]2

Rb a

Ry a(b−x)

α−1₍_b₋_y₎α−1₍_g₍_x₎₋_g₍_y₎₎2

×f(x)f(y)w(x)w(y)dxdy= 1

[Γ(α)]2

Rb a(b−y)

α−1

×f(y)w(y)Ry a(b−x)

α−1_g2₍_x₎_f₍_x₎_w₍_x₎_dxdy

+Rb a(b−y)

α−1_g2₍_y₎_f₍_y₎_w₍_y₎Ry a(b−x)

α−1

×f(x)w(x)dxdy−2Rb a(b−y)

α−1_g₍_y₎_f₍_y₎

×w(y)Ry a(b−x)

α−1_g₍_x₎_f₍_x₎_w₍_x₎_dxdy

.

(3.11)

Then, we can write

Rb a(b−y)

α−1_f₍_y₎_w₍_y₎Ry a(b−x)

α−1_g2₍_x₎

×f(x)w(x)dxdy=Rb a(b−y)

α−1_f₍_y₎_w₍_y₎_dy

Rb

a(b−x)

α−1_g2₍_x₎_f₍_x₎_w₍_x₎_dx₋Rb

a(b−y)

α−1

×g2(y)f(y)w(y)Ry a(b−x)

α−1_f₍_x₎_w₍_x₎_dxdy

(3.12)

and

Z b

a

(b−y)α−1g(y)f(y)w(y)

Z y

a

(b−x)α−1g(x)

×f(x)w(x)dxdy=

Z b

a

(b−y)α−1g(y)f(y)w(y)dy 2

−

Z b

a

(b−y)α−1g(y)f(y)w(y)

Z y

a

(b−x)α−1

×g(x)f(x)w(x)dxdy. (3.13)

By(3.11),(3.12)and(3.13),we obtain 1

[Γ(α)]2

Z b

a

Z y

a

(b−x)α−1(b−y)α−1(g(x)−g(y))2

×f(x)f(y)w(x)w(y)dxdy

= 1

[Γ(α)]2 Z _b

a

(b−y)α−1f(y)w(y)dy Z b

a

(b−x)α−1g2(x)

×f(x)w(x)dx−

Z b

a

(b−y)α−1g(y)f(y)w(y)dy 2#

= Jα

a f w(b)Jaαg2f w(b)−(Jaαg f w(b)) 2

= Jα

ag2f w(b)−(Jaαg f w(b)) 2

= E_g2_,_α_,_w−E_g2_,_α_,_w

= E_g2_,_α_,_w−2E_g2_,_α_,_w+E_g2_,_α_,_w

= E_g2_,_α_,_w−2E_g2_,_α_,_w+J_aαf w(b)E_g2_,_α_,_w

= E₍_g−Eg,α,w)2,α,w.

Finally, based on Lemma 3.7, we present to the reader the following results:

Theorem 3.9. Let X be a CRV with support an interval

[a,b],−∞<a<b<∞,having a pdf f and let w:[a,b]→R+

be a continuous function. Then, for anyα≥1, the following inequality holds

E_[_g₋_E

g,α,w]2,α,w≤Ez(g0)2

h0 ,α,w

, (3.14)

where g is an absolutely continuous function, withE_zg0_, α,ω

<

∞and J_aαw f(b) =1,h is continuous function on[a,b],with

0<h0and

z(t) = 1

(b−t)α−1_w₍_t₎_f₍_t₎ (3.15)

× t

Z

a

(b−u)α−1w(u)f(u)(Eh,α,w−h(u))du.

Proof:By Lemma 3.7, we have

E_[_g₋_E_g,α,w_]2_,_α_,_w

= 1

[Γ(α)]2

b

Z

a y

Z

a

(b−x)α−1(b−y)α−1[g(y)−g(x)]2

×f(x)w(x)f(y)w(y)dxdy

= 1

[Γ(α)]2

b

Z

a y

Z

a

(b−x)α−1(b−y)α−1





y

Z

x

g0(t)dt 



2

×f(x)w(x)f(y)w(y)dxdy.

(6)

Thanks to the condition onh0and by Cauchy-Schwarz inequality, we get





y

Z

x

g0(t)dt   2 =   y Z x p

h0₍_t₎ g 0₍_t₎

p h0(t)

! dt





2

≤ (h(y)−h(x))

y

Z

x

[g0(t)]2

h0(t) dt.(3.17)

By(3.16)and(3.17), we can write

E_[_g₋_E

g,α,w]2,α,w

≤ 1

[Γ(α)]2

b Z a y Z a

(b−x)α−1(b−y)α−1(h(y)−h(x))

× y

Z

x

[g0(t)]2

h0₍_t₎ dt f(x)w(x)f(y)w(y)dxdy.

On the other hand, since

 



x≤t≤y a≤x≤y a≤y≤b

⇔

 



a≤x≤t t≤y≤b a≤t≤b

,

then, by changing the order of integration, it yields that

E_[_g₋_E_g,α,w_]2_,_α_,_w

≤ 1

[Γ(α)]2

b

Z

a

[g0(t)]2

h0₍_t₎   b Z t t Z a

(b−x)α−1(b−y)α−1

×(h(y)−h(x))f(x)w(x)f(y)w(y)dxdy

dt.

(3.19)

Also, we have

b Z t t Z a

(b−x)α−1(b−y)α−1(h(y)−h(x))f(x)w(x)

×f(y)w(y)dxdy=

b Z a t Z a

(b−x)α−1(b−y)α−1

×(h(y)−h(x))f(x)w(x)f(y)w(y)dxdy

− t Z a t Z a

×f(y)w(y)dxdy

=

b

Z

a

(b−y)α−1h(y)f(y)w(y)dy

t

Z

a

(b−x)α−1f(x)w(x)dx

− b

Z

a

(b−y)α−1f(y)w(y)dy

t

Z

a

(b−x)α−1h(x)f(x)w(x)dx

− t

Z

a

t

Z

a

+

t

Z

a

t

Z

a

=

b

Z

a

t

Z

a

− b

Z

a

t

Z

a

(b−x)α−1h(x)f(x)w(x)dx.

(3.20)

Multiplying(3.20)by 1

Γ(α),we obtain 1 Γ(α) b Z t t Z a

×f(y)w(y)dxdy=Jα

a (h f w) (b) t

Z

a

(b−x)α−1f(x)

×w(x)dx−Jα

a(f w) (b) t

Z

a

= Jα

a (h f w) (b) t

Z

a

− t

Z

a

=

t

Z

a

(b−x)α−1[Jα

a(h f w) (b)−h(x)]f(x)w(x)dx

=

t

Z

a

(b−x)α−1

Eh,α,w−h(x)

f(x)w(x)dx. (3.21)

By (3.18) and (3.21), we get

E_[_g−Eg,α,w]2,α,w

≤ 1

Γ(α)

b

Z

a

[g0(t)]2

h0₍_t₎ 



t

Z

a

(b−x)α−1Eh,α,w−h(x)

f(x)w(x)dx  dt = 1 Γ(α) b Z a

[g0(t)]2

h0₍_t₎

(b−t)α−1w(t)f(t)z(t)dt

= Jα

a

_z₍_g0₎2

h0 w f

(7)

Theorem 3.9 is thus achieved.

Remark 3.10. If we take α=1,and w(x) =1in (3.14), we obtain (1.4).

References

[1] _{T. Cacoullos and V. Papathanasiou: On upper bounds}

for the variance of functions of random varialbes, Statist. Probab. Lett., 7 (1985), pp. 175-184.

[2] _{T. Cacoullos and V. Papathanasiou: Caracterizations of}

distributuons by variance bounds. Math. Statist., 4 (1989), pp. 351-356.

[3] _{T. Cacoullos and V. Papathanasiou: Characterizations of}

distributions by generalisations of variance bounds and simple proofs of the CLT, J. Statist. Plann. Inference, 63 (1997), pp. 157-171.

[4] _{Z. Dahmani, A.E. Bouziane, M. Houas and M.Z.}

Sarikaya: New w-weighted concepts for continuous ran-dom variables with applications, Note di Matematica, 37(1) (2017), pp. 23-40.

[5] _{Z. Dahmani: New applications of fractional calculus on}

probabilistic random variables, Acta Math. Univ. Come-nianaeVol. LXXXVI, 2 (2017), pp. 299-307.

[6] _{F. Goodarzi, M. Amini and G.R. Mohtashami Borzadaran:}

On upper bounds for the variance of functions of random variables with weighted distributions, Labachevskii Jour-nal of Mathematics, 37(4) (2016), pp. 422-435.

[7] _{R. Gorenflo, F. Mainardi: Fractional calculus: integral}

and differential equations of fractional order, Springer Verlag, Wien, (1997), pp. 223-276.

[8] _{G.R. Mohtashami Borzadaran and D.N. Shanbhag:}

Statist. Probab. Lett., 39 (1998), pp. 109-117.

[9] _{M.Z. Sarikaya, M.E. Kiris,N. Celik: Hermite-Hadamard}

type inequality for h-convex stochastic processes. AIP Conference Proceedings 1726, 020076 (2016).

[10] _{M.Z. Sarikaya, T. Tunc, H. Budak: Simpson’s type}

in-equality for F-convex function. FACTA Univ. FUMI, 1(10), (2018), pp. 747-753.

[11] _{M.Z. Sarikaya, H. Yildiz, H. Budak: On weighted}

Iyengar-type inequalities for conformable fractional inte-grals. Mathematical Sciences, 11(4), (2017), pp. 327-331.

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Further results and applications on continuous random variables