UNIT 3 MOMENTS 3.1 INTRODUCTION. Objectives. Structure 3.1 Introduction. 3.2 Introduction to Moments 3.3 Methods for Calculation of Moments

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1 UIT MOMETS Momets Stcte. Itocto Objectves. Itocto to Momets. Metos o Calclato o Momets Momets abot btay Pots Momets abot Og Momets abot Mea Relato betwee Momets abot Mea a Momets abot btay Pots Eect o Cage o Og a Scale o Momets Seppa s Coectos o Momets Cale s Cec o Momets. Peaso s Beta a Gamma Coecets.5 Smmay.6 Soltos / swes. ITRODUCTIO Momets ae poplaly se to escbe te caactestc o a stbto. Tey epeset a coveet a yg meto o smmazg may o te most commoly se statstcal meases sc as meases o teecy, vaato, sewess a toss. Momets ae statstcal meases tat gve ceta caactestcs o te stbto. Momets ca be aw momets, cetal momets a momets abot ay abtay pot. Fo eample, te st aw momet gves mea a te seco cetal momet gves vaace. ltog ect omlae est o cetal momets eve te tey ca be easly calclate wt te elp o aw momets. Te t cetal momet o te vaable s tmes te t cetal momet o wee ( )/ s a ew vaable obtae by sbjectg to a cage o og a scale. Sce oes ot come to te scee so tee s o eect o cage o og o momets. Te basc cocepts o momets ae escbe Secto.. I Secto., te metos o calclato o eet s o momets ae eploe. I ts secto te metos o calclato o aw momets, momets abot zeo a cetal momets ae eplae. I Secto., te Peaso s Beta a Gamma coecets o sewess a toss ae escbe. We sall scss te sewess a toss etals Ut o ts bloc. Objectves te styg ts t, yo wol be able to ee momets; epla eet types o momets a te ses; eve te elato betwee aw a cetal momets; 5

2 alyss o Qattatve Data escbe te eect o cage o og a scale o momets; calclate te aw a cetal momets o gope a gope eqecy stbtos; ee te Sepa s coecto o momets; a calclate te Beta a Gamma coecets o sewess a toss.. ITRODUCTIO TO MOMETS Momet wo s vey popla mecacal sceces. I scece momet s a mease o eegy wc geeates te eqecy. I Statstcs, momets ae te atmetc meas o st, seco, t a so o,.e. t powe o te evato tae om ete mea o a abtay pot o a stbto. I ote wos, momets ae statstcal meases tat gve ceta caactestcs o te stbto. I statstcs, some momets ae vey mpotat. Geeally, ay eqecy stbto, o momets ae obtae wc ae ow as st, seco, t a ot momets. Tese o momets escbe te omato abot mea, vaace, sewess a toss o a eqecy stbto. Calclato o momets gves some eates o a stbto wc ae o statstcal mpotace. Momets ca be classe aw a cetal momet. Raw momets ae mease abot ay abtay pot (say). I s tae to be zeo te aw momets ae calle momets abot og. We s tae to be tmetc mea we get cetal momets. Te st aw momet abot og s mea weeas te st cetal momet s zeo. Te seco aw a cetal momets ae mea sqae evato a vaace, espectvely. Te t a ot momets ae sel measg sewess a toss.. METHODS OF CLCULTIO OF MOMETS Tee types o momets ae:. Momets abot abtay pot,. Momets abot mea, a. Momets abot og.. Momets abot btay Pot We actal mea s acto, momets ae st calclate abot a abtay pot a te covete to momets abot te actal mea. We evatos ae tae om abtay pot, te omla s ae: Fo Ugope Data I,,..., ae te obsevatos o a vaable X, te te momets abot a abtay pot ae 5 Zeo oe momet ( )

3 Fst oe momet ( ) Momets Seco oe momet ( ) T oe momet ( ) a Fot oe momet ( ) I geeal te t oe momet abot abtay pot s gve by ( ) ; o,, Fo Gope Data I,,..., ae vales (o m vales case o class tevals) o a vaable X wt te coespog eqeces,,...,, te momets abot a abtay pot ae Zeo oe momet ( ) ; wee Fst oe momet Seco oe momet T oe momet ( ) ( ) ( ) Fot oe momet ( ) I geeal, te t oe momet abot a abtay pot ca be obtae as ( ) ;,, 55

4 alyss o Qattatve Data 56 I a eqecy stbto, to smply calclato we ca se sot-ct meto. I o te, we get te momets abot a abtay pot ae Fst oe momet Seco oe momet T oe momet Fot oe momets Smlaly, t oe momet abot s gve by ; o,,.. Momets abot Og I case, we we tae a abtay pot te, we get te momets abot og. Fo Ugope Data Te t oe momet abot og s ee as: t oe momet Fst oe momet Seco oe momet T oe momet Fot oe momet

5 Fo Gope Data Momets t oe momet ( Fst oe momet ) ( ) Seco oe momet ( ) T oe momet ( ) Fot oe momet.. Momets abot Mea We we tae te evato om te actal mea a calclate te momets, tese ae ow as momets abot mea o cetal momets. Te omlae ae: Fo Ugope Data Zeo oe momet Fst oe momet ( ) ( ) Ts st oe momet abot mea s zeo, becase te algebac sm o te evato om te mea s zeo Seco oe momet ( ). σ (vaace) Teeoe, seco oe momet abot mea s vaace. Tese eslts, vz,, a σ ae o vey mpotat statstcal teoy a pactcal. T oe momet ( ) 57

6 alyss o Qattatve Data a Fot oe momet ( ) I geeal, te t oe momet o a vaable abot te mea s gve by Fo Gope Data ( ) ; o,,, I case o eqecy stbto te t oe momet abot mea s gve by: ( ) ; o,, By sbstttg te eet vale o ae we ca gate eet oes momet abot mea as ollows: Zeo oe momet ( ) as Fst oe momet ( ) Becase Seco oe momet T oe momet ( ) ( ) Vaace (σ ) 58 a Fot oe momet ( ).. Relato betwee Momets abot Mea a Momets abot btay Pot Te t momet abot mea s gve by ( ) ( ) ;wee,,

7 59 Momets ) ( ) ( () I - te, + Teeoe, we get om eqato () ) ( ) ( ) (... ) ( C ) ( C C Te,... C C C () I patcla o pttg, a eqato (), we get 6..5 Eect o Cage o Og a Scale o Momets Let so tat a Ts, t momet o abot abtay pot s gve by ) ( ) ( ) (... C C C

8 alyss o Qattatve Data a, t momet o abot mea s gve by ( ) ( ) ( ) Ts, te t momet o te vaable abot mea s tmes te t momet o te ew vaable abot mea ate cagg te og a scale...6 Seppa s Coectos o Momets Te ametal assmpto tat we mae amg class tevals s tat te eqeces ae omly stbte abot te m pots o te class tevals. ll te momet calclatos case o gope eqecy stbtos ely o ts assmpto. Te aggegate o te obsevatos o te powes a class s appomate by mltplyg te class m pot o ts powe by te coespog class eqecy. Fo stbtos tat ae ete symmetcal o close to beg symmetcal, ts assmpto s acceptable. Bt t s ot acceptable o gly sewe stbtos o we te class tevals ecee abot / t o te age. I sc statos, W. F. Seppa sggeste some coectos to be mae to get o te so calle gopg eos tat ete to te calclato o momets. Seppa sggeste te ollowg coectos ow as Seppa s coectos te calclato o cetal momets assmg cotos eqecy stbtos te eqecy tapes o to zeo bot ectos ( coecte ) ( coecte ) 7 ( coecte ) wee, s te wt o class teval...7 Cale s Cecs o Momets Te ollowg ettes 6

9 Momets 6 ae se to cec te calclatos oe o g momets.. PERSO S BET D GMM COEFFICIETS Kal Peaso ee te ollowg o coecets, base po te st o cetal momets:. s ee as It s se as mease o sewess. Fo a symmetcal stbto, sall be zeo. as a mease o sewess oes ot tell abot te ecto o sewess,.e. postve o egatve. Becase beg te sm o cbes o te evatos om mea may be postve o egatve bt s always postve. lso beg te vaace always postve. Hece wol be always postve. Ts awbac s emove we calclate Kal Peaso s coecet o sewess wc s te sqae oot o,. e. γ β σ Te te sg o sewess wol epe po te vale o wete t s postve o egatve. It s avsable to se as mease o sewess.. meases toss a t s ee by β a smlaly, coecet o toss s ee as Eample : Fo te ollowg stbto calclate st o momets abot mea a also,, a : Mas Feqecy 6 6 6

10 alyss o Qattatve Data Solto: Fst we costct te ollowg eqecy stbto o calclato o momets: Mas () ( ) Te Momets abot mea (. 5.5) + 6 (.5) (.) (.)

11 Momets E) Te st o momets o a stbto abot te vale 5 o a vaable ae,, a 5. F te cetal momets, a. E) Fo te ollowg stbto, cetal momets, a : Class Feqecy 7 E) Wages o woes ae gve te ollowg table: Weely Wages Feqecy 7 F te st o cetal momet a a..5 SUMMRY I ts t, we ave scsse:. Wat s momets;. Te eet types o momets a te ses;. Te elato betwee aw a cetal momets;. Te eect o cage o og a scale o momets; 5. How to calclate te aw a cetal momets o te gve eqecy stbto; 6. Sepa s Coecto o Momets; a 7. Te Beta a Gamma coecets..6 SOLUTIOS / SWERS E) We ave gve,, a 5 ow we ave to ot momets abot mea 6

12 alyss o Qattatve Data So teeoe, β 79 8 β.7 E) Fo calclato o momets We teeoe ave, Sce tat meas te stbtos s symmetcal.

13 s, < tat meas cve s platytc. E) Calclato o st o momets Momets 7 Wages Teeoe, sg omla So, we ave ,

14 alyss o Qattatve Data Ts, te β coecets β a β