Document Details

SilentAntigorite9958

Uploaded by SilentAntigorite9958

University of North Carolina at Chapel Hill

Tags

binomial distribution probability statistics mathematics

Summary

These lecture notes cover the binomial distribution, including examples and calculations. The focus is on the probability of getting a specific number of successes in a set number of trials.

Full Transcript

Lecture 10 DCB 4e, Sec. 4.3 %LQRPLDOGLVWULEXWLRQ 6XSSRVHZHUDQGRPO\VHOHFWIRXULQGLYLGXDOVWRSDUWLFLSDWHLQWKLV H[SHULPHQW:KDWLVWKHSUREDELOLW\WKDWH[DFWO\RIWKHPZLOOUHIXVHWR DGPLQLVWHUWKHVKRFN" 6XSSRVHZHUDQGRPO\VHOHFWIRXULQGLYLGXDOVWRSDUWLFLSDWHLQWKLV H[SHULPHQW...

Lecture 10 DCB 4e, Sec. 4.3 %LQRPLDOGLVWULEXWLRQ 6XSSRVHZHUDQGRPO\VHOHFWIRXULQGLYLGXDOVWRSDUWLFLSDWHLQWKLV H[SHULPHQW:KDWLVWKHSUREDELOLW\WKDWH[DFWO\RIWKHPZLOOUHIXVHWR DGPLQLVWHUWKHVKRFN" 6XSSRVHZHUDQGRPO\VHOHFWIRXULQGLYLGXDOVWRSDUWLFLSDWHLQWKLV H[SHULPHQW:KDWLVWKHSUREDELOLW\WKDWH[DFWO\RIWKHPZLOOUHIXVHWR DGPLQLVWHUWKHVKRFN" /HW VFDOOWKHVHSHRSOH$OOHQ $ %ULWWDQ\ % &DUROLQH & DQG'DPLDQ '  (DFKRQHRIWKHIRXUVFHQDULRVEHORZZLOOVDWLVI\WKHFRQGLWLRQRI³H[DFWO\ RIWKHPUHIXVHVWRDGPLQLVWHUWKHVKRFN´ 6XSSRVHZHUDQGRPO\VHOHFWIRXULQGLYLGXDOVWRSDUWLFLSDWHLQWKLV H[SHULPHQW:KDWLVWKHSUREDELOLW\WKDWH[DFWO\RIWKHPZLOOUHIXVHWR DGPLQLVWHUWKHVKRFN" /HW VFDOOWKHVHSHRSOH$OOHQ $ %ULWWDQ\ % &DUROLQH & DQG'DPLDQ '  (DFKRQHRIWKHIRXUVFHQDULRVEHORZZLOOVDWLVI\WKHFRQGLWLRQRI³H[DFWO\ RIWKHPUHIXVHVWRDGPLQLVWHUWKHVKRFN´ 6XSSRVHZHUDQGRPO\VHOHFWIRXULQGLYLGXDOVWRSDUWLFLSDWHLQWKLV H[SHULPHQW:KDWLVWKHSUREDELOLW\WKDWH[DFWO\RIWKHPZLOOUHIXVHWR DGPLQLVWHUWKHVKRFN" /HW VFDOOWKHVHSHRSOH$OOHQ $ %ULWWDQ\ % &DUROLQH & DQG'DPLDQ '  (DFKRQHRIWKHIRXUVFHQDULRVEHORZZLOOVDWLVI\WKHFRQGLWLRQRI³H[DFWO\ RIWKHPUHIXVHVWRDGPLQLVWHUWKHVKRFN´ 6XSSRVHZHUDQGRPO\VHOHFWIRXULQGLYLGXDOVWRSDUWLFLSDWHLQWKLV H[SHULPHQW:KDWLVWKHSUREDELOLW\WKDWH[DFWO\RIWKHPZLOOUHIXVHWR DGPLQLVWHUWKHVKRFN" /HW VFDOOWKHVHSHRSOH$OOHQ $ %ULWWDQ\ % &DUROLQH & DQG'DPLDQ '  (DFKRQHRIWKHIRXUVFHQDULRVEHORZZLOOVDWLVI\WKHFRQGLWLRQRI³H[DFWO\ RIWKHPUHIXVHVWRDGPLQLVWHUWKHVKRFN´ 6XSSRVHZHUDQGRPO\VHOHFWIRXULQGLYLGXDOVWRSDUWLFLSDWHLQWKLV H[SHULPHQW:KDWLVWKHSUREDELOLW\WKDWH[DFWO\RIWKHPZLOOUHIXVHWR DGPLQLVWHUWKHVKRFN" /HW VFDOOWKHVHSHRSOH$OOHQ $ %ULWWDQ\ % &DUROLQH & DQG'DPLDQ '  (DFKRQHRIWKHIRXUVFHQDULRVEHORZZLOOVDWLVI\WKHFRQGLWLRQRI³H[DFWO\ RIWKHPUHIXVHVWRDGPLQLVWHUWKHVKRFN´ 6XSSRVHZHUDQGRPO\VHOHFWIRXULQGLYLGXDOVWRSDUWLFLSDWHLQWKLV H[SHULPHQW:KDWLVWKHSUREDELOLW\WKDWH[DFWO\RIWKHPZLOOUHIXVHWR DGPLQLVWHUWKHVKRFN" /HW VFDOOWKHVHSHRSOH$OOHQ $ %ULWWDQ\ % &DUROLQH & DQG'DPLDQ '  (DFKRQHRIWKHIRXUVFHQDULRVEHORZZLOOVDWLVI\WKHFRQGLWLRQRI³H[DFWO\ RIWKHPUHIXVHVWRDGPLQLVWHUWKHVKRFN´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Ɣ RIVFHQDULRVWKHUHLVDOHVVWHGLRXVZD\WRILJXUHWKLV RXWZH OOJHWWRWKDWVKRUWO\ %LQRPLDOGLVWULEXWLRQ 7KHTXHVWLRQIURPWKHSULRUVOLGHDVNHGIRUWKHSUREDELOLW\RI JLYHQQXPEHURIVXFFHVVHVNLQDJLYHQQXPEHURIWULDOVQ N VXFFHVVLQQ WULDOV DQGZHFDOFXODWHGWKLV SUREDELOLW\DV RIVFHQDULRV[3 VLQJOHVFHQDULR Ɣ RIVFHQDULRVWKHUHLVDOHVVWHGLRXVZD\WRILJXUHWKLV RXWZH OOJHWWRWKDWVKRUWO\ Ɣ 3 VLQJOHVFHQDULR  SN S QN ZKHUHSLVWKHSUREDELOLW\RIVXFFHVVWRWKHSRZHURIQXPEHURIVXFFHVVHVSUREDELOLW\RIIDLOXUH WRWKHSRZHURIQXPEHURIIDLOXUHV %LQRPLDOGLVWULEXWLRQ 7KHTXHVWLRQIURPWKHSULRUVOLGHDVNHGIRUWKHSUREDELOLW\RI JLYHQQXPEHURIVXFFHVVHVNLQDJLYHQQXPEHURIWULDOVQ N VXFFHVVLQQ WULDOV DQGZHFDOFXODWHGWKLV SUREDELOLW\DV RIVFHQDULRV[3 VLQJOHVFHQDULR Ɣ RIVFHQDULRVWKHUHLVDOHVVWHGLRXVZD\WRILJXUHWKLV RXWZH OOJHWWRWKDWVKRUWO\ Ɣ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he correct answer for (d) is n. %LQRPLDOGLVWULEXWLRQ FRQW %LQRPLDOSUREDELOLWLHV ,ISUHSUHVHQWVSUREDELOLW\RIVXFFHVV S UHSUHVHQWV SUREDELOLW\RIIDLOXUHQUHSUHVHQWVQXPEHURILQGHSHQGHQW WULDOVDQGNUHSUHVHQWVQXPEHURIVXFFHVVHV Binomial probability formula 3UDFWLFH :KLFKRIWKHIROORZLQJLVQRWDFRQGLWLRQWKDWQHHGVWREHPHW IRUWKHELQRPLDOGLVWULEXWLRQWREHDSSOLFDEOH" D WKHWULDOVPXVWEHLQGHSHQGHQW E WKHQXPEHURIWULDOVQPXVWEHIL[HG F HDFKWULDORXWFRPHPXVWEHFODVVLILHGDVDVXFFHVV RUDIDLOXUH G WKHQXPEHURIGHVLUHGVXFFHVVHVNPXVWEHJUHDWHUWKDQ WKHQXPEHURIWULDOV H WKHSUREDELOLW\RIVXFFHVVSPXVWEHWKHVDPHIRUHDFK WULDO 3UDFWLFH :KLFKRIWKHIROORZLQJLVQRWDFRQGLWLRQWKDWQHHGVWREHPHW IRUWKHELQRPLDOGLVWULEXWLRQWREHDSSOLFDEOH" D WKHWULDOVPXVWEHLQGHSHQGHQW E WKHQXPEHURIWULDOVQPXVWEHIL[HG F HDFKWULDORXWFRPHPXVWEHFODVVLILHGDVDVXFFHVV RUDIDLOXUH G WKHQXPEHURIGHVLUHGVXFFHVVHVNPXVWEHJUHDWHUWKDQ WKHQXPEHURIWULDOV H WKHSUREDELOLW\RIVXFFHVVSPXVWEHWKHVDPHIRUHDFK WULDO 3UDFWLFH $*DOOXSVXUYH\VXJJHVWVWKDWRI$PHULFDQVDUH REHVH$PRQJDUDQGRPVDPSOHRI$PHULFDQVZKDWLVWKH SUREDELOLW\WKDWH[DFWO\DUHREHVH" D SUHWW\KLJK E SUHWW\ORZ *DOOXSKWWSZZZJDOOXSFRPSROOREHVLW\UDWHVWDEOHDVS[-DQXDU\ 3UDFWLFH $*DOOXSVXUYH\VXJJHVWVWKDWRI$PHULFDQVDUH REHVH$PRQJDUDQGRPVDPSOHRI$PHULFDQVZKDWLVWKH SUREDELOLW\WKDWH[DFWO\DUHREHVH" D SUHWW\KLJK E SUHWW\ORZ *DOOXSKWWSZZZJDOOXSFRPSROOREHVLW\UDWHVWDEOHDVS[-DQXDU\ 3UDFWLFH $*DOOXSVXUYH\VXJJHVWVWKDWRI$PHULFDQVDUH REHVH$PRQJDUDQGRPVDPSOHRI$PHULFDQVZKDWLVWKH SUREDELOLW\WKDWH[DFWO\DUHREHVH" 3UDFWLFH $*DOOXSVXUYH\VXJJHVWVWKDWRI$PHULFDQVDUH REHVH$PRQJDUDQGRPVDPSOHRI$PHULFDQVZKDWLVWKH SUREDELOLW\WKDWH[DFWO\DUHREHVH" 7KHELUWKGD\SUREOHP :KDWLVWKHSUREDELOLW\WKDWUDQGRPO\FKRVHQSHRSOHVKDUHD ELUWKGD\" 7KHELUWKGD\SUREOHP :KDWLVWKHSUREDELOLW\WKDWUDQGRPO\FKRVHQSHRSOHVKDUHD ELUWKGD\" 3UHWW\ORZ§ 7KHELUWKGD\SUREOHP :KDWLVWKHSUREDELOLW\WKDWUDQGRPO\FKRVHQSHRSOHVKDUHD ELUWKGD\" 3UHWW\ORZ§ :KDWLVWKHSUREDELOLW\WKDWDWOHDVWSHRSOHRXWRISHRSOH VKDUHDELUWKGD\" 7KHELUWKGD\SUREOHP (not a binomial model) :KDWLVWKHSUREDELOLW\WKDWUDQGRPO\FKRVHQSHRSOHVKDUHD ELUWKGD\" 3UHWW\ORZ§ :KDWLVWKHSUREDELOLW\WKDWDWOHDVWSHRSOHRXWRISHRSOH VKDUHDELUWKGD\" ([DFWO\ ([FOXGLQJWKHSRVVLELOLW\RIDOHDS\HDUELUWKGD\ 7KHELUWKGD\SUREOHP FRQW :KDWLVWKHSUREDELOLW\WKDWDWOHDVWSHRSOH PDWFK RXWRI SHRSOHVKDUHDELUWKGD\" 7KHELUWKGD\SUREOHP FRQW :KDWLVWKHSUREDELOLW\WKDWDWOHDVWSHRSOH PDWFK RXWRI SHRSOHVKDUHDELUWKGD\" 6RPHZKDWFRPSOLFDWHGWRFDOFXODWHEXWZHFDQWKLQNRILWDVWKH FRPSOHPHQWRIWKHSUREDELOLW\WKDWWKHUHDUHQRPDWFKHVLQ SHRSOH 7KHELUWKGD\SUREOHP FRQW :KDWLVWKHSUREDELOLW\WKDWDWOHDVWSHRSOH PDWFK RXWRI SHRSOHVKDUHDELUWKGD\" 6RPHZKDWFRPSOLFDWHGWRFDOFXODWHEXWZHFDQWKLQNRILWDVWKH FRPSOHPHQWRIWKHSUREDELOLW\WKDWWKHUHDUHQRPDWFKHVLQ SHRSOH 7KHELUWKGD\SUREOHP FRQW :KDWLVWKHSUREDELOLW\WKDWDWOHDVWSHRSOH PDWFK RXWRI SHRSOHVKDUHDELUWKGD\" 6RPHZKDWFRPSOLFDWHGWRFDOFXODWHEXWZHFDQWKLQNRILWDVWKH FRPSOHPHQWRIWKHSUREDELOLW\WKDWWKHUHDUHQRPDWFKHVLQ SHRSOH 7KHELUWKGD\SUREOHP FRQW :KDWLVWKHSUREDELOLW\WKDWDWOHDVWSHRSOH PDWFK RXWRI SHRSOHVKDUHDELUWKGD\" 6RPHZKDWFRPSOLFDWHGWRFDOFXODWHEXWZHFDQWKLQNRILWDVWKH FRPSOHPHQWRIWKHSUREDELOLW\WKDWWKHUHDUHQRPDWFKHVLQ SHRSOH 7KHELUWKGD\SUREOHP FRQW :KDWLVWKHSUREDELOLW\WKDWDWOHDVWSHRSOH PDWFK RXWRI SHRSOHVKDUHDELUWKGD\" 6RPHZKDWFRPSOLFDWHGWRFDOFXODWHEXWZHFDQWKLQNRILWDVWKH FRPSOHPHQWRIWKHSUREDELOLW\WKDWWKHUHDUHQRPDWFKHVLQ SHRSOH 7KHELUWKGD\SUREOHP FRQW :KDWLVWKHSUREDELOLW\WKDWDWOHDVWSHRSOH PDWFK RXWRI SHRSOHVKDUHDELUWKGD\" 6RPHZKDWFRPSOLFDWHGWRFDOFXODWHEXWZHFDQWKLQNRILWDVWKH FRPSOHPHQWRIWKHSUREDELOLW\WKDWWKHUHDUHQRPDWFKHVLQ SHRSOH '' detour '' (why ?) ([SHFWHGYDOXH $*DOOXSVXUYH\VXJJHVWVWKDWRI$PHULFDQVDUH REHVH $PRQJDUDQGRPVDPSOHRI$PHULFDQVKRZPDQ\ZRXOG \RXH[SHFWWREHREHVH" ([SHFWHGYDOXH $*DOOXSVXUYH\VXJJHVWVWKDWRI$PHULFDQVDUH REHVH $PRQJDUDQGRPVDPSOHRI$PHULFDQVKRZPDQ\ZRXOG \RXH[SHFWWREHREHVH" Ɣ (DV\HQRXJK[  ([SHFWHGYDOXH $*DOOXSVXUYH\VXJJHVWVWKDWRI$PHULFDQVDUH REHVH $PRQJDUDQGRPVDPSOHRI$PHULFDQVKRZPDQ\ZRXOG \RXH[SHFWWREHREHVH" Ɣ (DV\HQRXJK[  Ɣ 2UPRUHIRUPDOO\— QS [  ([SHFWHGYDOXH $*DOOXSVXUYH\VXJJHVWVWKDWRI$PHULFDQVDUH REHVH $PRQJDUDQGRPVDPSOHRI$PHULFDQVKRZPDQ\ZRXOG \RXH[SHFWWREHREHVH" Ɣ (DV\HQRXJK[  Ɣ 2UPRUHIRUPDOO\— QS [  Ɣ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“ [ ĺ  3UDFWLFH $Q$XJXVW*DOOXSSROOVXJJHVWVWKDWRI$PHULFDQV WKLQNKRPHVFKRROLQJSURYLGHVDQH[FHOOHQWHGXFDWLRQIRU FKLOGUHQ:RXOGDUDQGRPVDPSOHRI$PHULFDQVZKHUH RQO\VKDUHWKLVRSLQLRQEHFRQVLGHUHGXQXVXDO"  D 

Use Quizgecko on...
Browser
Browser