SAMPLE MOMENTS. POPULATION MOMENTS.. Momets about the org raw momets. The rth momet about the org of a radom varable X, deoted by µ r, s the expected value of X r ; symbolcally, µ r EX r x x r fx for r 0,,,... whe X s dscrete ad µ r EX r x r fx dx whe X s cotuous. The rth momet about the org s oly defed f E[ X r exsts. A momet about the org s sometmes called a raw momet. Note that µ EX µ X, the mea of the dstrbuto of X, or smply the mea of X. The rth momet s sometmes wrtte as fucto of θ where θ s a vector of parameters that characterze the dstrbuto of X. If there s a sequece of radom varables, X,X,...X, we wll call the rth populato momet of the th radom varable µ, r ad defe t as µ,r E X r 3.. Cetral momets. The rth momet about the mea of a radom varable X, deoted by µ r, s the expected value of X µ X r symbolcally, µ r E[X µ X r x x µ X r fx 4 for r 0,,,... whe X s dscrete ad µ r E[X µ X r x µ X r fx dx whe X s cotuous. The r th momet about the mea s oly defed f E[ X - µ X r exsts. The rth momet about the mea of a radom varable X s sometmes called the rth cetral momet of X. The rth cetral momet of X about a s defed as E[ X - a r. If a µ X, we have the rth cetral momet of X about µ X. Note that 5 Date: December 7, 005.
SAMPLE MOMENTS µ E[X µ X µ E[X µ X x µ X fx dx 0 x µ X fx dx VarX σ 6 Also ote that all odd momets of X aroud ts mea are zero for symmetrcal dstrbutos, provded such momets exst. If there s a sequece of radom varables, X,X,... X, we wll call the r th cetral populato momet of the th radom varable µ,r ad defe t as µ,r E X r r, µ 7 Whe the varables are detcally dstrbuted, we wll drop the subscrpt ad wrte µ r ad µ r.. SAMPLE MOMENTS.. Deftos. Assume there s a sequece of radom varables, X,X,...X. The frst sample momet, usually called the average s defed by X X 8 Correspodg to ths statstc s ts umercal value, x, whch s defed by x x 9 where x represets the observed value of X. The rth sample momet for ay t s defed by X r Ths too has a umercal couterpart gve by X r 0 x r x r.. Propertes of Sample Momets.... Expected value of X r. Takg the expected value of equato 0 we obta E [ X r E X r If the X s are detcally dstrbuted, the EX r µ,r E [ X r E X r µ r µ r 3
SAMPLE MOMENTS 3... Varace of X r. Frst cosder the case where we have a sample X,X,...,X. Var X r Var If the X s are depedet, the X r Var X r 4 Var X r VarX r 5 If the X s are depedet ad detcally dstrbuted, the Var X r VarXr 6 where X deotes ay oe of the radom varables because they are all detcal. I the case where r, we obta Var X Var X σ 7 3. SAMPLE CENTRAL MOMENTS 3.. Deftos. Assume there s a sequece of radom varables, X,X,...,X. We defe the sample cetral momets as C r C C These are oly defed f µ, s kow. X µ r,,r,, 3,..., X µ, X µ, 8 3.. Propertes of Sample Cetral Momets. 3... Expected value of C r. The expected value of C r s gve by E C r The last equalty follows from equato 7. E X µ r, µ,r 9 If the X are detcally dstrbuted, the E C r µ r E C 0 0
4 SAMPLE MOMENTS 3... Varace of C r. Frst cosder the case where we have a sample X,X,...,X. Var C r Var X µ, If the X s are depedetly dstrbuted, the Var C r If the X s are depedet ad detcally dstrbuted, the r Var Var [ X µ r, X µ r, Var C r Var[ X µ r 3 where X deotes ay oe of the radom varables because they are all detcal. I the case where r, we obta Var C Var[ X µ Var[ X µ σ Cov[ X, µ+var[ µ 4 σ 4. SAMPLE ABOUT THE AVERAGE 4.. Deftos. Assume there s a sequece of radom varables, X,X,...X. Defe the rth sample momet about the average as m r M r X X r, r,, 3,..., 5 Ths s clearly a statstc of whch we ca compute a umercal value. We deote the umercal value by,, ad defe t as I the specal case where r we have m r x x r 6 M X X X X X X 0 7 4.. Propertes of Sample Momets about the Average whe r.
SAMPLE MOMENTS 5 4... Alteratve ways to wrte M r. We ca wrte M a alteratve useful way by expadg the squared term ad the smplfyg as follows M r r X X M X X [ X X X X + X X X + X 8 X X + X X X 4... Expected value of M. The expected value of M r s the gve by E [ M E X E [ X E [ X [ E X Var X µ, µ, Var X 9 The secod le follows from the alteratve defto of varace Var X E X [ E X E X [ E X + Var X 30 E X [ E X + Var X ad the thrd le follows from equato. If the X are depedet ad detcally dstrbuted, the
6 SAMPLE MOMENTS E [ M E X E [ X µ, µ, Var X µ µ σ 3 σ σ σ where µ ad µ are the frst ad secod populato momets, ad µ s the secod cetral populato momet for the detcally dstrbuted varables. Note that ths obvously mples [ E X X E M σ σ 3 4..3. Varace of M. By defto, Var M [ M E EM 33 The secod term o the rght o equato 33 s easly obtaed by squarg the result equato 3. E M σ E M EM σ 4 34 Now cosder the frst term o the rght had sde of equato 33. Wrte t as [ M E E X X 35 Now cosder wrtg X X as follows
SAMPLE MOMENTS 7 X X X µ X µ Y Ȳ 36 where Y X µ Ȳ X µ Obvously, X X Y Ȳ, where Y X µ,ȳ X µ 37 Now cosder the propertes of the radom varable Y whch s a trasformato of X. Frst the expected value. Y X µ E Y E X E µ µ µ 0 38 The varace of Y s Y X µ VarY VarX σ f X are depedetly ad detcally dstrbuted 39 Also cosder EY 4. We ca wrte ths as E Y 4 y 4 f x dx x µ 4 fx dx 40 µ 4 Now wrte equato 35 as follows
8 SAMPLE MOMENTS [ M E E E E X X X X Y Ȳ E Y Ȳ 4a 4b 4c 4d Igorg for ow, expad equato 4 as follows E Y Ȳ E Y Y Ȳ + Ȳ E { E { E E E Y Ȳ Y Y Y Y Y + } Ȳ + Ȳ } Ȳ Ȳ [ E Ȳ Y Ȳ + Ȳ 4 Y 4a 4b 4c 4d 4e + E Ȳ 4 4f Now cosder the frst term o the rght of 4 whch we ca wrte as
SAMPLE MOMENTS 9 E Y E E [ EY 4 Y j Yj Y 4 + Y j Yj µ 4 + µ µ 4 + σ 4 + j EY EYj 43a 43b 43c 43d 43e Now cosder the secod term o the rght of 4 gorg for ow whch we ca wrte as E [ Ȳ Y E Y j Y k Y j k 44a E Y 4 + Y Yj + Y jy k j j k j k Y 44b EY 4 + j EY EYj + EY jey k j k j k EY 44c [ µ4 + µ +0 44d [ µ4 + σ 4 44e The last term o the peultmate le s zero because EY j EY k EY 0.
0 SAMPLE MOMENTS Now cosder the thrd term o the rght sde of 4 gorg for ow whch we ca wrte as E [ Ȳ 4 E 4 E Y j Y j k Y k l Y l Y 4 + Y Yk + Y Yj + k j j Y Yj + 45a 45b where for the frst double sum j kl, for the secod k jl, ad for the last l j k ad... dcates that all other terms clude Y a o-squared form, the expected value of whch wll be zero. Gve that the Y are depedetly ad detcally dstrbuted, the expected value of each of the double sums s the same, whch gves E [ Ȳ 4 4 E 4 [ 4 [ Y 4 + Y Yk + Y Yj + k j j EY 4 +3 Y Yj + terms cotag EX j EY 4 +3 Y Yj j Y Y j + 46a 46b 46c 4 [ µ4 +3 µ 46d 4 [ µ4 +3 σ 4 46e 3 [ µ4 +3 σ 4 46f Now combg the formato equatos 44, 45, ad 46 we obta
SAMPLE MOMENTS E Y Ȳ E Y Y Ȳ + Ȳ E Y [ E Ȳ Y 47a + E Ȳ 4 47b [ µ 4 + µ [ µ4 + µ [ + [ µ4 3 +3µ 47c µ 4 + µ [ µ 4 [ + µ [ + µ4 + 3 µ 47d µ 4 µ 4 + µ 4 + µ µ 3 + µ 47e + µ 4 + +3 µ 47f + µ 4 + +3 σ 4 47g Now rewrte equato 4 cludg as follows [ E M E Y Ȳ + µ 4 + +3 σ 4 48a 48b + 3 µ 4 + +3 3 σ 4 48c 3 µ 4 + +3 3 σ 4 48d Now substtute equatos 34 ad 48 to equato 33 to obta Var M [ M E EM 3 µ 4 + +3 3 σ 4 σ 4 49 We ca smplfy ths as
SAMPLE MOMENTS Var M [ M E EM 50a µ 3 4 + +3 σ 4 σ 4 50b 3 3 µ 4 + [ σ 4 + 3 50c µ 4 + [ σ 4 + 3 + 3 3 µ 4 + [ σ 4 3 3 µ 4 [ σ 4 3 3 µ 4 3 3σ4 3 50d 50e 50f 5. SAMPLE VARIANCE 5.. Defto of sample varace. The sample varace s defed as S X X 5 We ca wrte ths terms of momets about the mea as S X X M where M 5 X X 5.. Expected value of S. We ca compute the expected value of S by substtutg from equato 3 as follows E S E M σ σ 53 5.3. Varace of S. We ca compute the varace of S by substtutg from equato 50 as follows
SAMPLE MOMENTS 3 Var S Var M µ 4 3 3 σ4 3 54 µ 4 3σ4 5.4. Defto of ˆσ. Oe possble estmate of the populato varace s ˆσ whch s gve by ˆσ X X M 55 5.5. Expected value of ˆσ. We ca compute the expected value of ˆσ by substtutg from equato 3 as follows E ˆσ E M σ 56 5.6. Varace of ˆσ. We ca compute the varace of ˆσ by substtutg from equato 50 as follows Var ˆσ Var M µ 4 3 µ 4 µ We ca also wrte ths a alteratve fasho Var ˆσ Var M 3 σ4 3 µ 4 µ + µ 4 3µ 3 57 µ 4 3 3σ4 3 µ 4 3µ 3 3 µ 4 µ 4 + µ 4 3 µ 4 µ +3µ 3 58 µ 4 µ µ 4 µ +µ 4 3 µ 3 µ 4 µ µ 4 µ + µ 4 3 µ 3
4 SAMPLE MOMENTS 6. NORMAL POPULATIONS 6.. Cetral momets of the ormal dstrbuto. For a ormal populato we ca obta the cetral momets by dfferetatg the momet geeratg fucto. The momet geeratg fucto for the cetral momets s as follows M X t e t σ. 59 The momets are the as follows. The frst cetral momet s EX µ d dt e t σ tσ e t σ t 0 t 0 60 0 The secod cetral momet s EX µ d e t σ dt d dt t 0 t σ e t σ t σ 4 e t σ t 0 + σ e t σ t 0 6 σ The thrd cetral momet s EX µ 3 d3 dt 3 e t σ d dt t 0 t σ 4 e t σ t 3 σ 6 e t σ t 3 σ 6 e t σ + σ e t σ +tσ 4 e t σ +3tσ 4 e t σ t 0 + tσ 4 e t σ t 0 t 0 6 0 The fourth cetral momet s
SAMPLE MOMENTS 5 EX µ 4 d4 dt 4 e t σ d dt t 0 t 3 σ 6 e t σ t 4 σ 8 e t σ t 4 σ 8 e t σ +3tσ 4 e t σ +3t σ 6 e t σ +6t σ 6 e t σ t 0 +3t σ 6 e t σ +3σ 4 e t σ +3σ 4 e t σ t 0 t 0 63 3σ 4 6.. Varace of S. Let X,X,...X be a radom sample from a ormal populato wth mea µ ad varace σ <. We kow from equato 54 that Var S Var M µ 4 3 3 σ4 3 64 µ 4 3σ4 If we substtute for µ 4 from equato 63 we obta Var S µ 4 3σ4 3 σ4 3σ4 3 3σ4 3 3 +3σ4 65 σ4 6.3. Varace of ˆσ. It s easy to show that σ4 Varˆσ σ4 66