DSpace at VNU: On the mean absolute deviation of the random variables

VN U , J O U R N A L OF SCIENCE (Mat, Sci., J X V n®5 - 1999 ON TH E M E A N A BSO L U T E D EV IA TIO N OF T H E R A N D O M V A R IA BLES T n Loc H u n g Department of Matiieiimtic College o f Sciences Universitv o f Hue P h a m G ia T h u De pm tĩnent o f Mãtheiìiíìtic mid Statistics University of Moiictoii, Moncton, C huhcI h A b s t r a c t : The m ain object of the study is a measure o f dispersion, is nam ed the Mean Absolute Deviation (or A4AD, fo r skort) o f a random variable X, ỗf^{X) = E{\ X - \) The basic properties of the M A D and som,e detailed computations on t,he M A D are established We also focus Oil the applications of the M A D in the Limit Theorems, when the role of the stmidard devĩaỉìon Ơ^ { X) — [ E { X — is p l a y e d by Ỗ ^ { X ) IN T R O D U C T IO N Let X be a random variable with finite mean E ( X ) — ỊI T he stan d a rd deviation of X, denoted by Ơ^ { X) — [ i ’ivV “ is very weil-kiiown in tlie probabilistic and statistical literature as a moasure of (ỉispeisioii Especially, its wide-spread use has presented in theory of limit theoienis, ill sampling thooiy ill th e analysis of variaiico aiul statistical decision theory (see [1], [2], [3], [6] an(i [71 for coniploto bibliography) O n the u l h f i n d , althuu^l i phiVtij^ a dumiiiaiil luh* Hi fuiKtioiial analvMii, Tiic mean absolute deviation (or MAD, for short) of X (li'notrd by seen relatively few applications ill probability iiiul s ta tis tic s a{ X) = E(\ X — ụ I), lias In tho traditional terniinolo^\' is said to be the firsi absolute Iiioinont of a rantloin vai'iabl(‘ X {sec [1], [2j [.'i^ Hiul 8] for the definition) Probably, their coinputaiioual complexities are not convenit'ut to use, especially when the random variables are discrete (see for instance Section 2) However, from the inequality < ơị ^{X) for an a rb itra ry random variable X (see Proposition 2.5), the question arises as to w hat happens if the role of the stauclard deviation Ơ^ { X) is played by In recent years some results concerning th e M AD have been investigated by P ham Gia THU, Q p DUON G and Ttukan N in some topics of statistics, econometrics, reliability theory and Bayesian analysis (see [4], [5] and [6] for more details) T he main aim of this note is to present the basic properties of the M AD of a rand om v a r ia b le a b o u t its m e a n ổ ^ ị X ) a n d a p p l i c a t i o n s in t h e lim it t h e o r e m s , w h e n t h e ro le o f 36 O n the m e a n a b so lu te d e v i a t i o n o f the 37 t h e s t a n d a n l d e v i a t i o n f ^ { X ) is plavtnl by Mo i( ‘ specificallv, in Sort ion and SOI I K' i l l u s t r a t i v o co n ip u tatio n s W(‘ li'view iiu MADs S 01 K' a i(‘ also o f main pi-OỊ)ii E { X ) — IA aiul íli(’ staiulai(! (l('\‘iaTioii iy( A ') — A lo obfaiii that ^ ị(.Y ) = ' I (.r - / ■ A ‘ )Aj-’r' '"•’ilr = r ' X ' = 2 i .r > I’o > (3 Then where /i = E { X ) = Proof It can he verified that fi = E { X ) = r ^ a r i;r -'',l:r = ^ r o I , n - Taking (1) m to account W(‘ g('t Ồ, AX) - / ”“ ' ■'h, - 2.ro(o - 1) - ‘ (1 - a - ' '' The pioof is sti ai^lii-foi ward Ộ P r o p o s i t i o n 2.7, Let X bt: a Poissoii disirilxiicd Ỉ(ui(ỉ()ifi (uiruihif’ (Ijilit fhe po.sftn'i ừdeger-vaỉue mean E { X ) n n > Thtii 0„{ X) ^ n c - " ' ^ ~ J ị v = c j „ { X ) J ị ^ d 8 „ { X ) fi] \ ' 7Ĩ V 7T where the sụpt ^ ÌÌ —^ -j-oc is used to liidicafc that thf! lafio of Ỉỉic t(V0 sỉdcs tffids fo (ui/tiỊ (LS Proof N o te tha t, \\ip varianci' a^^(A') also is the positive int(‘gri-valu(‘ // By (liiectly using iiw fonnula (2) from Lem m a 2.1 wo will show th at „ ( X) = y (n -k )c -"~ = h'l nr~"-^ //! O n the m e a n absohite d e v i a ti o n o f the,, Using tho Stirling's foiiiiula //! = \ / ^ n " sions), W(* ^(’t Ỉ , (s(H* [2; p 50] for (ỉ('taiỉ('(l tliscĩỊS- ;/! w h i U P t h e siftii ỈÌ —» + 0 Ộ For tlu' is u s e d t o i n d i c a t e t h a t case, wo not(' th at as E t h e r a t i o o f t h e t w o s i d e s t t ’i i d s t o uriit.y a s 11—» + - V- MI „ (X ) I 00, /2 V 7T ’ where Ị-1 = E [ X ) is the mean of X anil ơ' Ì {X) is the vaiiancp of X P r o p o s i t i o n (see [2, piohloin 35 p 226 Lei S ,1 be the number o f success in n Bernoulli trials with the inejiii E{ S n) = 1>P and the variance ơịj,{S„) = ĩìpq,{ũ < p < l , p + q = 1) Then ỏ„,,(5„) = E (| 5,, - up ‘i n v q _ ^ - \J Proof A (liii'ct coiiipiitation iVoiii the t'onnula (2) of Lemma 2.1 shows th a t ["/'] 0„,,{S„) = E(\ 5„ - np I) = ‘l Y ^ i n p - k ) c y < r ^ = 2Av;C*//V/' A-U \vh(‘ỉ(' k is tlif integer Iiuiubor such th at iip < Ả' < iip + 1b y (‘Oiitiiiuity usinp, again th(' Stirling, s toiniula, tor sutticienMy largo ri, wo have = E{\ s„ - lip I) ~ = \Ị^ „ ị,(S „ ) This concludes the proof, ộ T he sam e conclusion can be drawn for this case, as n —♦ + 0 , OẢX) where /i =: E { X ) is tlie mean of X and ị { X ) is the variance of X P r o p o s i t i o n 2.9 Let X be a normal distributed random vaTiable with the mean E { X ) = Ị1 and the variance V n r { X ) = ị { X ) Then Tran Loc Hung, P h a m G i a Thu Proof By using (1) from L e m m a 2.1 , ail oasv coinputation shows that = /" " (.r - 0.79788rT,(A-) rrv/27T V In t h e s a n u ' m a u n o i a s al)OV(> W(‘ c a n SCO TĨ tliat, I A ’ - /T 7T as was to be shown LIM IT T H E O R E M S From above Piopơsitioiis 2.8 and 2.7 we can now pK'sent rliP following ipsults T h e o r e m 3.1 Lei A 'l, A'2 be a sequence of idenUcaUy independent brno- mtal difitrrhufed random variables witti tin: inenv.s Eị Xị , ) = p {{) < 1) niiđ the vuTimice VariXi^-) = pq.yk- = n Set s„ - A'a- rhtrn s„ - E( S „ )_ ^ E ơ( S„) s„ '■2 lip 7Ĩ where the sign ~ is used to riidtcafe that ihe ratio of the two sides tends to unity as V —* + 00 Thts gives Proof X I a( S„) ' — (IS n — It is asily M'CII that S,I !)(' a Iiimilx'i OỈ success t/1 11 first Bernoulli trials witli E( S „) = up ami V(i r(S„) -= Iijxi- W(> now a[){)lv arguiiK'ui as in Proposition 2,8 ai>aiii, with X K 'plaa’d by s„ to o b tain coinplctc proof, T h e o r e m 3.2 LeJ X , X X , he a sequence of identically independent Pois- son drsfri.buted random variiihles wtth the Jiieaiis E{Xf , ) = A, (A G ^ ) and the vartav.cc Vari Xf ) = A,VA- = , n Set s „ = E I '=1 s„ E - E{S „ ) (y{S„) Then „\S„ - n \ vAÃ where tht siyn ~ If, usr.d to indicate that the ratio o f the two sides tends to unrty as ÌÌ -hcx) Tììis shows that i n j S n - nX\ \ / o ^ i V ‘- A/ ^A On the m e a n absolute d e v i a ti o n o f the I t l o l l o w s i i m u < ‘( l i a ( ’l\' t h a t l^ o f pnraiiH'tci iiX X > 0, 13 S'„ 1)C a l í u i í l í ^ i i i \ a i i a l ) l t ' (ji i l n ‘ l a w w i t h tilt' ) — / / A, I ' d f i S t , ) = / / A A i i a l \ s i s s i n i i l a i 1() r l i a t ill t l i c Ị ) i o o i ' o f P i o Ị ì o s i r i o i i , w i t h X I ( ‘Ị ) l a c ( ' ( l 1>\’ , v „ \V(‘ c a n f i n i s h t l i i ’ ] n ( ) ( Ộ Foi'iu IU)\V \V(' will í o n n u l a t( ' K'siilts c o i u e i n i i i ^ thí' W('ak laws lai‘í^(' Iimubcis wIk'h tlu* l o l v of ílií' s t a n d a r d (li‘\'iatiuii ~ Noti' t h a t t I k ’ (ulluvviu^ re su lt s arc íh(' laws of Hii* plav(‘il by o f tlii* \v } < -^ ,(A ') Pt'ooj T h ( ‘ (3) is has('(l on th(* following ul)S('rvatiuii for all f > () I ll(' Ị)l UOt \ -Ỉ'- if ' r i u ' o r e m ‘d li Ị I \ Ộ (riio \\Vak Law uf Lar^c Nuinbois for a i h i t i a r v laiuloin vaiial)l('s): I j ỉ A | A ) ()C a sr(ỊUtrní'(‘ o f ì d c n t i c a ỉ l ị Ị ì V ì ì d o n ì lUỊruỉ hỉ cs (