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Exam 1 Solutions

# Exam 1 Solutions - Hohfwulfdo dqg.rpsxwhu Hqjlqhhulqj...

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Hohfwulfdo dqg rpsxwhu Hqjlqhhulqj/ Lqvwuxfwru= Gu1 Rnhfkxnzx 1 Xjzhmh 7733=78:288:= Zluhohvv rppxqlfdwlrqv H[DP &4 Vsulqj 5335 Iheuxdu| 44/ 5335 Sureohp 4= +43 . 43 swv1, d , Jlyhq ehorz duh wkh ±uvw ulqj ri d fhooxodu dufklwhfwxuh1 Ghwhuplqh wkh uhxvh glvwdqfh G lq whupv ri wkh udglxv U 1 Vlqfh T @ [email protected] @ I 6 Q> zh kdyh wkh iroorzlqj= l , G @ I 6 ± 7 U @ 5 I 6 U @ 6 = 79 U ll , G @ I 6 ± : U @ I 54 U @ 7 = 8; U lll , G @ I 6 ± < U @ 6 I 6 U @ 8 = 5 U ly , G @ I 6 ± 45 U @ I 69 U @ 9 U y , G @ I 6 ± 46 U @ I 6< U @ 9 = 58 U yl , G @ I 6 ± 49 U @ 7 I 6 U @ 9 = <6 U e , hoo vhfwrulqj lv xvhg lq wkh hooxodu vwuxfwxuh vkrzq1 Ghwhuplqh wkh Uhxvh Iuhtxhqf| hoov iru wkh lqglfdwhg vhfwru1 15 6 13 20 2 4 3 7 5 19 21 12 14 17 10 18 11 16 9 8 1 Assume N = 7 Sohdvh vhh Ohfwxuh Sduw LLL/ sdjh 751 Sureohp 5= +:.:.9 swv1, Lq d hooxodu whohskrqh v|vwhp/ wkh qxpehu ri fdoov shu fxvwrphu shu exv| krxu lv :8( 1 Dvvxph wkdw Q @ 7 / eorfnlqj suredelolw| lv 6( / kroglqj wlph lv 433 vhf1/ dqg r²huhg ordg shu fhoo lv :: = : Huodqjv1 Vhh Ohfwxuh Sduw LLL/ sdjhv 69/ 6:1 4

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d , Zkdw lv wkh qxpehu ri fdoov shu fhoo shu krxuB Qr1 ri fdoo2fhoo2ku1 @ :: = : ± 6933 vhf 433 vhf @ 5:<: = 5 e , Zkdw lv wkh qxpehu ri fxvwrphuv shu fhooB Qr1 ri fxvwrphuv2fhoo @ Qr1 ri fdoov2fhoo2ku Qr1 ri fdoov2fxvwrphu2ku1 @ 5:<: = 5 3 = :8 @ 6:63 f , Zkdw lv wkh qxpehu ri fdoov shu fkdqqho shu krxuB Qr1 ri fdoov2fkdqqho2ku1 @ Qr1 ri fdoov2fhoo2ku Dydlodeoh fkdqqho @ 5:<: = 5 +666 ² 54, @ 7 @ 68 = ;9 Sureohp 6= +9.9.; swv1, D 83 p wudqvplw wrzhu frppxqlfdwhv wr d preloh ryhu yhu| frqgxfwlyh jurxqg1 Wkh fduulhu lv Krul}rqwdo Srodul}hg dw d iuhtxhqf| ri 933 PK}1 Wkh preloh lv 8 np iurp wkh wudqvplwwhu dqg wkh dqwhqqd lv orfdwhg 4 = 8 p deryh wkh jurxqg1 d , Zkdw lv wkh iuhh vsdfh orvvB O EL @ 43 orj 43 ± ± 7 ²g ² 5 @ 53 orj 43 ± +633 @ 933, 7 ² ± 8333 ² @ 434 = <; gE @ ² 435 gE e , Zkdw lv wkh dgglwlrqdo orvv gxh wr pxowlsdwkB O FI @ 43 orj 43 ± 7 ²k M k K ±g ² 5 @ 53 orj 43 7 ²k M k K ±g @ ; = 7: gE f , Li wkh pd{lpxp dffhswdeoh sdwk orvv lv ² 455 = 8 gE/ zkdw lv wkh pd{lpxp udqjhB g C- @ 53 orj 43 k M . 53 orj 43 k K . 455 = 8 @ 493 g @ 43 + C ( 73, @ 43 +493 ( 73, @ 43 np Sureohp 7= +9.9.; swv1, Dvvxph d uhfhlyhu lv orfdwhg 43 np iurp d 83 Z wudqvplwwhu1 Wkh fduulhu iuhtxhqf| lv <33 PK}1 Iuhh vsdfh sursdjdwlrq lv dvvxphg zlwk J 9 @ 4 dqg J 7 @ 5 1 ± @ f i @ 6 ± 43 ; [email protected] <33 ± 43 9 K} @ 3 = 666 d , Zkdw lv wkh uhfhlyhg srzhuB S 7 @ S 9 J 9 J 7 ± 5 +7 ²g , 5 @ 83 ± 5 ± 4 ± ± 3 = 666 7 ± ² ± 43333 ² 5 @ : = 3 ± 43 ± 43 Z @ ² <4 = 88 gEZ e , Ilqg wkh h²hfwlyh dshuwxuh ri wkh uhfhlyhu dqwhqqd1 D @ J 7 ± 5 7 ² @ 5 ± +3 = 666, 5 7 ² @ 3 = 34:98 p 5 f , Dvvxplqj wkdw wkh khljkw ri wkh uhfhlyhu lv 8 p/ zkdw lv wkh khljkw ri wkh dqwhqqdB Iurp S 7 @ S 9 J 9 J 7 k 5 M k 5 K g 7 5
zh jhw k M @ L S 7 g 7 S 9 J 9 J 7 k 5 K @ g 5 k K K S 7 S 9 J 9 J 7 @ +43333, 5 8 K : = 3 ± 43 ± 43 83 ± 4 ± 5 @ 85 = <5 p Sureohp 8= +53 swv1, d ,

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Exam 1 Solutions - Hohfwulfdo dqg.rpsxwhu Hqjlqhhulqj...

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