Homework 6 - Solution

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Shawn Wade
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1 Howork 6 - oluo 364: 79 Rfr o xal 7 Th aou of fll ss by a bolg ach s orally srbu wh σ= ouc If = 9 bols ar raoly slc fro h ouu of h ach w fou ha h robably ha h sal a wll b wh 3 ouc of h ru a s 638 uos ha s o b cou usg a sal of sz a If = 6 wha s P μ 3? b F P μ 3 wh s o b cou usg sals of szs = 5 = 36 = 49 a = 64 c ha ar o you obsrv aog h valus for P μ 3 ha you obsrv for h varous valus of? Do h rsuls ha you oba ar b s o b coss wh h rsul oba xal 73? oluo: By salg srbuo Nμ σ a N0 whr σ = a h = P P P P 7698 b P P P P 3 3 For = a 64 h robabls ar a 9836 rscvly c Th robabls cras wh whch s uv sc h varac of crass wh s hs rsuls ar coss sc h robably was lss ha 95 for valus of lss ha : 70 Rfr o xrcs 79 Assu ow ha h aou of fll ss by h bolg ach s orally srbu wh σ = oucs a If = 9 bols ar raoly slc fro h ouu of h ach wha s P μ 3? Coar hs wh h aswr oba xal 7 b F P μ 3 wh s o b cou usg sals of szs = 5 = 36 = 49 a = 64 c ha ar o you obsrv aog h valus for P μ 3 ha you obsrv for h varous valus of? How o h rscv robabls oba hs robl whr σ = coar o hos oba xrcs 79 whr σ =? oluo:

2 P 3 P 5 5 P 5 a P 3 P 5 P I s sallr ha wha w go xal 7 b For = 5 P 3 P ; For = 36 P 3 P ; For = 49 P 3 P ; For = 64 P 3 P c Th robabls crass wh Th robabls ar sallr wh a largr saar vao or ffus sy 375: 749 Th lgh of rqur for h roc aac of a auoobl or aohr ach usually has a ou-sha robably srbuo Bcaus so occasoal log srvc s wll occur h srbuo s o b skw o h rgh uos ha h lgh of rqur o ru a 5000-l chck a o srvc a auoobl has a 4 hours a saar vao 7 hour uos also ha h srvc ar las o srvc 50 auoobls r 8-hour ay a ha orr o o so ca s a axu avrag srvc of oly 6 hours r auoobl O wha rooro of all workays wll h srvc ar hav o work ovr? oluo: By salg srbuo Nμ σ Thus 64 P 6 P P : 770 I hs sco w rov h rul of hub ha h oral aroxao o h boal srbuo s aqua f 3 q ls h rval ha s f 0 3 q a 3 q a how ha 3 q f a oly f 9 q b how ha 0 3 q f a oly f 9 q c Cob h rsuls fro ars a a b o oba ha h oral aroxao o h boal s aqua f q a 9 9 q

3 or quvally 9 largr of a q sallr of a q oluo: a 3 q 3 q 3 q q 9 q q 9 q b 3 q 0 3 q 9 q 9 q c Pars a a b ly ha > 9 axq q If q axq q = q; ohrws axq q = q Tha s o say ha h largr of a q always gos o h o whl h sallr gos o h boo Thus largr of a q 9 sallr of a q xra ou hav robably har of h oral aroxao o boal srbuo Ths aroxao ca b a rcs usg covrgc srbuo Ths robl asks you o cosr Thor a r h su of Thor f h xac srbuo of jusfy your rsuls b Prov Thor usg a alcao of h CLT c Prov Thor usg covrgc of o grag fucos oluo: a Th o grag fuco of s c ar Th o grag fuco of s b By CLT w hav Thus B 0 whr a

4 o h CLT ca b rwr as c L whr Thus L h By Th 65 Thus w hav By Taylor s Thor 0 whr 0 0 Fro abov w kow ha a c 0 a so 0 a Thus 0 0 As 0 quz Thor a 0 ξ Thrfor as x 0 0 0

5 By covrgc of o grag fucos N0 Thus N N0 xra Thr s also a Posso aroxao o h boal srbuo s Thor a Prov Thor ; b Challg Ca you rov a hyscal rrao of hs rsul? hr woul w s sohg lk a boal srbuo wh a vry larg a a vry sall oluo: a x as Thus w say ha covrgs srbuo o b Fro ar a w kow ha for larg a hus sall h boal srbuo woul aroach o Posso srbuo I ohr wors Posso srbuo s h xr cas of h boal srbuo wh s larg a s sall A xal woul b h oal ubr of occurrc of a rar sas a cra ara larg a sall

Term Structure of Interest Rates: The Theories

Term Structure of Interest Rates: The Theories Handou 03 Econ 333 Abdul Munasb Trm Srucur of Inrs Ras: Th Thors Trm Srucur Facs Lookng a Fgur, w obsrv wo rm srucur facs Fac : Inrs ras for dffrn maurs nd o mov oghr ovr m Fac : Ylds on shor-rm bond mor