Zadania na twierdzenia graniczne Zadanie 1 Z partii towaru o

Zadania na twierdzenia graniczne Zadanie 1 Z partii towaru o

=DGDQLDQDWZLHUG]HQLDJUDQLF]QH
=DGDQLH
=SDUWLLWRZDUXRZDGOLZR FLSREUDQRSUyE HOHPHQWRZ 2EOLF]\üSUDZGRSRGRELH VWZR HOLF]EDZDGOLZ\FKHOHPHQWyZZSUyELHQLHE G]LHSU]HNUDF]Dü
=DGDQLH
a) -DNGX DSRZLQQDE\üOLF]ED$DE\]SUDZGRSRGRELH VWZHPPR QDE\áRVWZLHUG]Lü
HZ UyGSREUDQ\FKORVRZRV]WXN]SDUWLLRZDGOLZR FLOLF]EDV]WXN ZDGOLZ\FK
nie przekroczy A?
b) -DN QDOH \SRVWDZLüKLSRWH] MH OLZ UyGORVRZRSREUDQ\FKV]WXNOLF]EDZDGOLZ\FK
SU]HNUDF]DZDUWR ü$"
=DGDQLH
:\VáDQRELWyZSU]H]PRGHPRNWyU\PZLDGRPR HFKDUDNWHU\]XMHVL VSUDZQR FL
WUDQVPLVML V\JQDáX = MDNLP SUDZGRSRGRELH VWZHP PR HP\ V G]Lü H Z UyG RZ\FK V\JQDáyZMHVWFRQDMZ\ HMEá GQ\FK2EOLF]HQLDSU]HSURZDG]LüNRU]\VWDM F
a) ]QLHUyZQR FL&]HE\V]HZD
b) z twierdzenie Moivre’a-Laplace’a
=DGDQLH
3DUWLD WRZDUX PD ZDGOLZR ü ,OX HOHPHQWRZ SUyE QDOH \ SREUDü DE\ ] SUDZGRSRGoELH VWZHPPR QDE\áRVWZLHUG]Lü HOLF]EDV]WXNZDGOLZ\FKZSUyELHE G]LH]DZDUWDZ
granicach od 2% do 4%?
=DGDQLH
3DUWLDWRZDUXPDZDGOLZR ü3REUDQRSUyE HOHPHQWRZ 2EOLF]\üSUDZGRSRGRELH VWZR HOLF]EDV]WXNZDGOLZ\FKZWHMSUyELHMHVW]DZDUWDZJUanicach od 6% do 9%?
=DGDQLH
: UyG DUyZHN SURGXNRZDQ\FK SU]H] SHZLHQ ]DNáDG MHVW EUDNyZ ,OH DUyZHN QDOH \
SREUDü DE\ ] SUDZGRSRGRELH VWZHP PR QD E\áR WZLHUG]Lü H E G]LHP\ PLHOL ZL FHM
QL DUyZHNZDGOLZ\FK"
=DGDQLH
=PLHQQD ORVRZD ; SU]\MPXMH ZDUWR FL UyZQH VXPLH OLF]E\ RF]HN SU]\ U]XFDQLX NRVWN GR
JU\.RU]\VWDM F]QLHUyZQR FL&]HE\V]HZDRV]DFRZDüSUDZGRSRGRELH VWZR3_;_!
:\QLNSRUyZQDü]GRNáDGQ ZDUWR FL WHJRSUDZGRSRGRELH VWZD
=DGDQLH
5]XFDP\PRQHW -DNZLHONDZLQQDE\üOLF]EDU]XWyZQDE\SUDZGRSRGRELH VWZRSVSHáQLeN
N
QLD QLHUyZQR FL - 0.5 < 0.1 , gdzie MHVW F] VWR FL Z]JO GQ Z\U]XFRQ\FK RUáyZ E\áR
Q
Q
ZL NV]HQL "1DOH \VNRU]\VWDü]QLHUyZQR FL&]HE\V]HZD
=DGDQLH
à F]\P\VL UD]\]LQWHUQHWHPNRU]\VWDM F]PRGHPXSU]\F]\PSUDZGRSRGRELH VWZR
X]\VNDQLD SRá F]HQLD ]D ND G\P UD]HP MHVW UyZQH 2NUH OLü SUDZGRSRGRELH VWZR H
OLF]EDX]\VNDQ\FK SRá F]H E G]LH VL Uy QLüRGQDMEDUG]LHM SUDZGRSRGREQHM LFK OLF]E\ QLH
ZL FHMQL RJyOQHMOLF]E\á F]H 6NRU]\VWDü]QLHUyZQR FL&]HE\V]HZD
=DGDQLH
:DULDQFMD ND GHM ] QLH]DOH Q\FK ]PLHQQ\FK ORVRZ\FK R MHGQDNRZ\P UR]NáDG]LH MHVW
UyZQD2EOLF]\üSUDZGRSRGRELH VWZR H UHGQLDW\FK]PLHQQ\FKRGFK\OLVL RGMHMZDUWoFLRF]HNLZDQHMQLHZL FHMQL R
=DGDQLH
=PLHQQDORVRZD<MHVW UHGQL DU\WPHW\F]Q QLH]DOH Q\FK]PLHQQ\FKORVRZ\FKRMHdQDNRZ\PUR]NáDG]LH]ZDUWR FL RF]HNLZDQ LZDULDQFM 2EOLF]\üSUDZGRSRGRELH VWZR
H<SU]\MPLHZDUWR ü]SU]HG]LDáX
=DGDQLH
=PLHQQD ORVRZD < MHVW UHGQL DU\WPHW\F]Q QLH]DOH Q\FK ]PLHQQ\FK ORVRZ\FK R MHGQDNoZ\PUR]NáDG]LHDZDULDQFMDND GHJRVNáDGQLNDZ\QRVL,OHQDOH \Z]L üVNáDGQLNyZDE\
<]SUDZGRSRGRELH VWZHPFRQDMPQLHMQLHRGFK\ODáDVL RGVZHMZDUWR FLRF]HNLZaQHMRZL FHMQL R"
=DGDQLH
5]XFDP\UD]\NRVWN GRJU\=QDOH üJUDQLFHZNWyU\FK]SUDZGRSRGRELH VWZHP
E G]LH]DZLHUDüVL á F]QDOLF]EDRF]HN
=DGDQLH
3U]\G]LDáDQLXSHZQHJRSURJUDPXNRPSXWHURZHJRQDOH \GRGDüOLF]E]NWyU\FKND GD
MHVW GDQD ] GRNáDGQR FL -m =DNáDGDM F H Eá G\ ]DRNU JOH V Z]DMHPQLH QLH]DOH QH L
PDM UR]NáDG UyZQRPLHUQ\ Z SU]HG]LDOH ¢10-m, 0.5¢10-m ]QDOH ü JUDQLFH Z NWyU\FK ]
SUDZGRSRGRELH VWZHPZL NV]\PQL ]DZLHUDüVL E G]LHá F]Q\UR]NáDGVXP\
=DGDQLH
3DUWLD]DZLHUDE EUDNyZ-DNLHMHVWSUDZGRSRGRELH VWZRS HZSUyELHOLF] FHMQ N
sztuk stosunek JG]LHNOLF]QDEUDNyZRGFK\OLVL RGERPQLHMQL "7RVDPR]DGDQLH
Q
QDOH \UR]ZL ]DüGODQ RUD]Q &RG]LHMHVL ]SUDZGRSRGRELH VWZHPSJG\OLFzEDQZ]UDVWD"-DNLHJRWZLHUG]HQLDLOXVWUDFM VWDQRZL SRZ\ V]HWU]\SU]\SDGNL"6NRU]\VWDM]
QLHUyZQR FL&]HE\V]HZD
=DGDQLH
3UDZGRSRGRELH VWZRZ\SURGXNRZDQLDGHWDOXSLHUZV]HMMDNR FLZSHZQ\P]DNáDG]LHZ\QRVL
2EOLF]\üSUDZGRSRGRELH VWZRWHJR HZ UyGORVRZRZ]L W\FKGHWDOLE G]LHGeWDOLSLHUZV]HMMDNR FL
=DGDQLH
5]XFDP\V\PHWU\F]Q PRQHW =QDOH üSUDZGRSRGRELH VWZR HRU]HáZ\SDGQLHUD]\
=DGDQLH
-DNLHMHVWSUDZGRSRGRELH VWZRZ\VW SLHQLDUD]\]MDZLVND$ ZVHULLGR ZLDGF]H MH HOLGR ZLDGF]HQLDV QLH]DOH QHLSUDZGRSRGRELH VWZR]MDZLVND$Z\QRVL
=DGDQLH
:RNUHVLHKGRNRQDQRORVRZRá F]H PRGHPX]OLQL WHOHIRQLF]Q -DNLHMHVWSUDZGoSRGRELH VWZR HZSU]HG]LDOHF]DVXOLF]EDá F]H ]PLH FLVL PL G]\L"
2GSRZLHG]L
1. 0.9065
2. a) A –EZDGOLZR ü!
3. a) p>1/3 b) p – 0.9887
4. n = 1118
5. 0.8532
6. ok. 1485
7. p ÊGRNáDGQDZDUWR üZ\QRVLS MDNRSUDZGRS]GDU]HQLDQLHPR Oiwego
8. n > 250
9. p Ë 0.938
10. 0.77
11. 0.97590
12. 450000
13. 3500  140
14. – ¡3/2¢10-m+2
15. dla n=100 p Ë 0.96, dla n=400 p Ë 0.9900, dla n=1600 p Ë 0.9975
16. tw. lokalnego Moivre’ a-Laplace’ a: j(-1.02)/¡24 = 0.04385
17. j/w: j(2)/50 = 0.00108
18. j/w: j(0)/7.05 = 0.0566
19. tw. globalne Moivre’ a-Laplace’ a: P(40 Ê k Ê 60) – f(2) - f(-2) = 0.9544