CHAPTER 6. d. With success = observation greater than 10, x = # of successes = 4, and
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1 CHAPTR 6 Secto 6.. a. We use the samle mea, to estmate the oulato mea µ. Σ 9.80 µ ~ 7. b. We use the samle meda, 7 (the mddle observato whe arraged ascedg order. c. We use the samle stadard devato, s s ( d. Wth success observato greater tha 0, # of successes 4, ad e. We use the samle (std dev/(mea, or. 09 s a. Wth # of T s the samle, the estmator s ; 0, so, b. Here, # samle wthout TI grahg calculator, ad 6, so. 80 0
2 Chater 6: Pot stmato. a. We use the samle mea,. 48 b. Because we assume ormalty, the mea meda, so we also use the samle mea.48. We could also easly use the samle meda. c. We use the 90 th ercetle of the samle: µ (.8.8s d. Sce we ca assume ormalty, P. s ( ( ( <. P Z < P Z < P( Z < e. The estmated stadard error of s 6 4. Y Y µ ; y a. ( ( ( µ b. V ( Y V ( V ( Y V Y s ( Y ; Y The estmate would be s s Y 7 0 s.660 c s.04 d. V ( Y V ( V ( Y N,000 T,76,00 y d 4. 0 N (,000(40.6,70,000 T Nd,76,00 (,000(4.0,9, T,76,00 y 74.6,60,
3 Chater 6: Pot stmato 6. a. Let of the y l( for I,..,. It s easly verfed that the samle mea ad samle sd y ' s are y. 0 ad s.496. Usg the samle mea ad samle sd y to estmate µ ad, resectvely, gves µ. 0 ad. 496 s.46. y (whece b. ( e µ µ ad ths eresso:. It s atural to estmate ( by usg µ ad.46 ( e.0 e( lace of a. µ 0. 6 b. τ 0, 000 µ,06, 000 c. 8 of 0 houses the samle used at least 00 therms (the successes, so d. The ordered samle values are 89, 99, 0, 09, 8,,, 8, 47, 6, from 8 whch the two mddle values are 8 ad, so µ 0. 0 ~^ ~ 8. a. Wth q deotg the true roorto of defectve comoets, (# defectve.. samle q.0 samle. sze 80 b. P(system works 68, so a estmate of ths robablty s
4 Chater 6: Pot stmato 9. a. ( µ ( λ, so s a ubased estmator for the Posso arameter λ ; ( 0(8 ((7... (7( 7, sce 0, 7. 0 λ. b. λ λ., so the estmated stadard error s a. ( Var( [ ( ] µ thus teds to overestmate µ., so the bas of the estmator s ; b. ( ks ( k( S µ k ( ks µ., so wth k,. ( (. a. ( ( b. Var q Var q Var q ( (, root of ths quatty. q Var( Var( ad the stadard error s the square c. Wth, q q q.,, q, the estmated stadard error s 7 76 d. (
5 Chater 6: Pot stmato 09 e (.880(.0 00 (.6(.6. ( ( ( ( ( ( S S S S ( (.. ( 6 4 ( d ( ( ( ( ( 4. a. m( 0 ad ma(, so the estmate of the umber of laes maufactured s ma( - m( 0 4. b. The estmate wll equal the true umber of laes maufactured ff m( α ad ma( β,.e., ff the smallest seral umber the oulato ad the largest seral umber the oulato both aear the samle. The estmator s ot ubased. Ths s because ma( ever overestmates β ad wll usually uderestmate t ( uless ma( β, so that [ma( ] < β. Smlarly, [m( ] > α,so [ma( - m( ] < β - α ; The estmate wll usually be smaller tha β - α, ad ca ever eceed t.. a. ( mles that. Cosder. The ( (, mlyg that s a ubased estmator for. b , so
6 Chater 6: Pot stmato 6. a. [ δ ( δ Y ] δ( ( δ ( Y δµ ( δ µ µ δ 4( δ δ Var( ( δ Var( Y m δ 8( δ Settg the dervatve wth resect to δ equal to 0 yelds 0 m 4m from whch δ. 4 m b. Var[ δ ( δ Y ]., 7. r r ( 0 r ( r! r r r 0!( r! 0 a. r ( ( ( b( ; r, b. For the gve sequece,, so a. b. ( µ f ( ; µ, e, so f ( µ ; µ, ad π π π π π ; sce, 4[[ f ( µ ] 4 > Var ( ~ > Var(. f ( µ π, so Var ~.467 ( π. 4 0
7 Chater 6: Pot stmato 9. λ, so λ. ad Y λ..; 0 the estmate s a... λ. b. ( ( λ. ( λ. λ., as desred. c. Here λ. 7 (.(., so 0 9 λ 7 70 ad 0 Y Secto a. We wsh to take the dervatve of ( d d for. l l ( ( l ( zero ad solvg for yelds l. For 0 ad,. 0, set t equal to zero ad solve ; settg ths equal to ; thus s a ubased estmator of. b. ( ( ( c. (.. 447
8 Chater 6: Pot stmato. β Γ ad ( Var ( [ ( ] β Γ α α momet estmators α ad β are the soluto to β Γ α, β Γ. Thus β α Γ α Γ s evaluated ad β the comuted. Sce β Γ α α Γ α, so ths equato must be solved to obta α. Γ α a. (, so the, so oce α has bee determed, b. From a, 0 6, α Γ α Γ α, so from the ht,. α. The β. Γ (. Γ(. Γ α Γ α, ad. d, so the momet estmator s the 0, yeldg. Sce. 80,. a. ( ( soluto to b. ( ( ( f,..., ;... d l ( l (. Takg d l( l( yelds ultmately.., so the log lkelhood s, so ad equatg to 0 yelds. Takg ( l for each gve
9 Chater 6: Pot stmato. For a sgle samle from a Posso dstrbuto, f (,..., ; λ λ e λ! λ e λ...! λ e λ!...!, so [ f (,..., ; ] l ( l (! λ λ λ. Thus [ [ ( ] f,..., ; λ 0 λ l d l. For our roblem, dλ,...,, y... y ; λ, λ λ ( λ f s a roduct of the samle lkelhood ad the y samle lkelhood, mlyg that ( ^ y λ. λ, λ y, ad (by the varace rcle r r 4. We wsh to take the dervatve of ( to zero, ad solve for : d d Settg ths equal to zero ad solvg for yelds r r l r l( l(. r. Ths s the umber of r l wth resect to, set t equal successes over the total umber of trals, whch s the same estmator for the bomal eercse 6.0. The ubased estmator from eercse 6.7 s same as the mamum lkelhood estmator. r r, whch s ot the. a. 84.4; s 9. 6 µ, so ( ( ad (ths s ot s. 9 0 b. The 9 th ercetle s µ. 64, so the mle of ths s (by the varace rcle µ The mle of P( 400 s (by the varace rcle 400 µ Φ Φ Φ 8.86 (
10 Chater 6: Pot stmato 7. a. f ( ; α, β (... α Σ / β e,...,, so the log lkelhood s α β Γ ( α ( α l( α l( β l Γ( α β 0 yelds l ( l ( β Γ( α 0. quatg both d α ad 0 dα β β dffcult system of equatos to solve. d d ad dα dβ, a very b. From the secod equato a, α αβ µ, so the mle of µ s µ. β to 8. a. e[ / ]... e[ / ] (... [ / ] e Σ atural log of the lkelhood fucto s l (... l ( Σ dervatve wrt ad equatg to 0 gves 0 Σ. The mle s therefore Σ estmator suggested ercse. b. For > 0 the cdf of f F( P(, so Σ Σ. The. Takg the ad, whch s detcal to the ubased ; s equal to e ths to. ad solvg for gves the meda terms of : that l (. ( e, so µ ~ The mle of µ ~ s therefore. quatg mles 4
11 Chater 6: Pot stmato 9. a. The jot df (lkelhood fucto s λσ( f λ e 0,..., ( ; λ,,..., Notce that,..., otherwse m, ff ( Σ λσ ad that λ ( λ. λ e( λσ e ( λ m ( Thus lkelhood 0 m ( < Cosder mamzato wrt. Because the eoet λ s ostve, creasg wll crease the lkelhood rovded that m ( ; f we make larger tha m m (, the lkelhood dros to 0. Ths mles that the mle of s ( The log lkelhood s ow l( λ λσ( ad solvg yelds λ.. quatg the dervatve wrt λ to 0 Σ Σ (. 0 ad Σ. 80, so λ b. m (.64, y;, where y y y 0. The lkelhood s ( ( f 4 λ 4λ ( 4 λe d P e rcle e 4λ 0 y λ y [ l ( ] We kow y for 0, y., so by the varace Sulemetary ercses µ ε µ ε. ( ( ( P µ > ε P µ > ε P µ < ε P > P < / / / / ε ε ε / z / z / P Z > P Z < e dz ε e dz. / π π As, both tegrals 0 sce z / lm e dz 0. c c π
12 Chater 6: Pot stmato. s a. F ( y P( Y y P( y y P( y... P( y Y for,..., y 0 y, so f Y ( y. y ( Y y dy Whle Y s ot ubased, Y 0 Y ( Y, so K b.. does the trck. y s, sce. Let the tme utl the frst brth, the elased tme betwee the frst ad secod brths, f,..., λ λ λ ; λ λe λ e... λ e! λσk k ad so o. The ( ( (. Thus d l! l λ λσk. Takg ad equatg to 0 yelds dλ λ. For the gve samle, 6,., , 9., 4 the log lkelhood s ( ( k k k k 6 4., 4.0, 6.; so k ((. ((6.... (6( ad λ. k k λ e 4. MS ( KS Var( KS Bas ( KS. Bas( KS ( KS K K (, ad ( [ ] [ ] ( ( S ( S K 4 Var ( KS K Var( S K K k the result s K 4 ( estmator (K or the mle (. To fd the mmzg value of K, take dk d ad equate to 0; ; thus the estmator whch mmzes MS s ether the ubased K. 6
13 Chater 6: Pot stmato. j There are averages, so the meda s the 8 th order of creasg magtude. Therefore, µ Wth. 86 ad, 490, s s ~ ' s are, creasg order,.0,.0,.08,.,.,.4,.,.4,.6,.8,.9,.,.7,.0,.4,.4,.7,.,.9,.0. The meda of these values s ( The. The estmate based o the resstat estmator s the. 7. Ths estmate s reasoably close agreemet wth s Let Γ( c Γ( Γ( 9. square root (S, leavg just. Whe 0, c. ( 0 9! Γ( 0 9 Γ ( 9. (8.(7...(.(. Γ(., but Γ( π gves c.0.. The (cs c(s, ad c cacels wth the two Γ factors ad the Γ ad.. Straghtforward calculato 7
14 Chater 6: Pot stmato 8. a. The lkelhood s ( ( ( ( µ µ µ µ y Σ Σ y Π e e e π π π ( Σ( µ ( µ lkelhood s thus ( Σ y zero gves lkelhood gves ( d l π. Takg dµ y µ. Substtutg these estmates of the s ' ( y y l π y d l ( π ( Σ( y. Now takg solvg for gves the desred result. d. The log ad equatg to µ to the log, equatg to zero, ad b. ( Σ( Y 4 Y ( Σ( Y 4, but ( V ( Y [ ( Y ] 0 ( Σ( 4 4. Thus, so the mle s deftely ot ubased; the eected value of the estmator s oly half the value of what s beg estmated! 8
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