Bi 3Cc phn phi xc sut thng gp

Phn phi nh thcPhp th BernoulliXt mt th nghim ch c 2 kh nng xy ra: thnh cng hoc tht bi.Thnh cng vi xc sut p.Tht bi vi xc sut 1-p. Th nghim nh vy gi l php th Bernoulli, k hiu B(1,p).

Phn phi nh thcPhp th Bernoulli v d.Tung ng xu: hnh / s.Mua v s: trng / khng trng.Tr li ngu nhin 1 cu trc nghim: ng / sai.Kim tra ngu nhin hng ha: tt / xu.

Phn phi nh thcPhn phi nh thcThc hin php th Bernoulli B(1,p) n ln c lp. tX = S ln thnh cng trong n ln th nghimX = 0, 1, 2, , n.X c phn phi nh thc vi tham s p.K hiu: X ~ B(n,p).

Phn phi nh thcCng thcXt X ~ B(n,p)

Phn phi nh thcV dCho X ~ B(5,0.1)Tnh P(X=1)

Phn phi nh thcHnh dng ca phn phi nh thc s ph thuc vo p v n.Meann = 5 v P = 0.1n = 5 v P = 0.5

Phn phi nh thcNu X ~ B(n,p):1) Trung bnh2) Phng sai v lch tiu chun n: s ln thc hin th nghim - p: xc sut thnh cng 1 ln th nghim- q = 1- p.

Phn phi nh thcn = 5 P = 0.1n = 5 P = 0.5Mean 0.2.4.6012345xP(x).2.4.6012345xP(x)0V d

Phn phi PoissonS cc bin c xy ra trong mt khong thi gian cho trc.S cc bin c trung bnh trn mt n v l .V dS ngi xp hng tnh tin siu th, s cuc in thoi n bu in trong 1 ngy, s my tnh h trong 1 ngy 1 khu vc,

Phn phi PoissonBin ngu nhin X nhn gi tr t 0, 1, 2, gi l c phn phi Poisson vi tham s nu

k = 0, 1, 2,

Phn phi PoissonTrung bnh

Phng sai v lch tiu chunVi = s bin c xy ra trung bnh trn 1 n v

Phn phi PoissonV dTrong mt nh my dt, bit s ng si b t trong 1 gi c phn phi Poisson vi trung bnh l 4. Tnh xc sut trong 1 gi ca. ng 3 ng si b t.b. C nhiu hn 1 ng si b t.

Bng tra phn phi PoissonV d: Tm P(X = 2) nu = .50

X0.100.200.300.400.500.600.700.800.90012345670.90480.09050.00450.00020.00000.00000.00000.00000.81870.16370.01640.00110.00010.00000.00000.00000.74080.22220.03330.00330.00030.00000.00000.00000.67030.26810.05360.00720.00070.00010.00000.00000.60650.30330.07580.01260.00160.00020.00000.00000.54880.32930.09880.01980.00300.00040.00000.00000.49660.34760.12170.02840.00500.00070.00010.00000.44930.35950.14380.03830.00770.00120.00020.00000.40660.36590.16470.04940.01110.00200.00030.0000

Phn phi xc sut PoissonP(X = 2) = .0758 = .50

X =0.50012345670.60650.30330.07580.01260.00160.00020.00000.0000

Chart2

0.6065306597

0.3032653299

0.0758163325

0.0126360554

0.0015795069

0.0001579507

0.0000131626

0.0000009402

x

P(x)

Histogram

0

0.6065306597

0.3032653299

0.0758163325

0.0126360554

0.0015795069

0.0001579507

0.0000131626

0.0000009402

0.0000000588

0.0000000033

0.0000000002

0

0

0

0

0

0

0

0

0

0

Number of Successes

P(X)

Histogram

Poisson2

Poisson Probabilities for Customer Arrivals

Data

Average/Expected number of successes:0.5

Poisson Probabilities Table

XP(X)P(=X)

00.6065310.6065310.0000000.3934691.000000

10.3032650.9097960.6065310.0902040.393469

20.0758160.9856120.9097960.0143880.090204

30.0126360.9982480.9856120.0017520.014388

40.0015800.9998280.9982480.0001720.001752

50.0001580.9999860.9998280.0000140.000172

60.0000130.9999990.9999860.0000010.000014

70.0000011.0000000.9999990.0000000.000001

80.0000001.0000001.0000000.0000000.000000

90.0000001.0000001.0000000.0000000.000000

100.0000001.0000001.0000000.0000000.000000

110.0000001.0000001.0000000.0000000.000000

120.0000001.0000001.0000000.0000000.000000

130.0000001.0000001.0000000.0000000.000000

140.0000001.0000001.0000000.0000000.000000

150.0000001.0000001.0000000.0000000.000000

160.0000001.0000001.0000000.0000000.000000

170.0000001.0000001.0000000.0000000.000000

180.0000001.0000001.0000000.0000000.000000

190.0000001.0000001.0000000.0000000.000000

200.0000001.0000001.0000000.0000000.000000

&A

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Poisson2

0

0

0

0

0

0

0

0

x

P(x)

Poisson

Poisson Probabilities for Customer Arrivals

Data

Average/Expected number of successes:0.1

Poisson Probabilities Table

XP(X)P(=X)

00.90480.9048370.0000000.0951631.000000

10.09050.9953210.9048370.0046790.095163

20.00450.9998450.9953210.0001550.004679

30.00020.9999960.9998450.0000040.000155

40.00001.0000000.9999960.0000000.000004

50.00001.0000001.0000000.0000000.000000

60.00001.0000001.0000000.0000000.000000

70.00001.0000001.0000000.0000000.000000

&A

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Sheet1

Sheet2

Sheet3

Phn phi PoissonHnh dng ca phn phi Poisson ph thuc vo tham s : =0.50 =3.00

Chart3

0.0497870684

0.1493612051

0.2240418077

0.2240418077

0.1680313557

0.1008188134

0.0504094067

0.0216040315

0.0081015118

0.0027005039

0.0008101512

0.0002209503

x

P(x)

Histogram

0

0.6065306597

0.3032653299

0.0758163325

0.0126360554

0.0015795069

0.0001579507

0.0000131626

0.0000009402

0.0000000588

0.0000000033

0.0000000002

0

0

0

0

0

0

0

0

0

0

Number of Successes

P(X)

Histogram

Poisson2

Poisson Probabilities for Customer Arrivals

Data

Average/Expected number of successes:3

Poisson Probabilities Table

XP(X)P(=X)

00.0497870.0497870.0000000.9502131.000000

10.1493610.1991480.0497870.8008520.950213

20.2240420.4231900.1991480.5768100.800852

30.2240420.6472320.4231900.3527680.576810

40.1680310.8152630.6472320.1847370.352768

50.1008190.9160820.8152630.0839180.184737

60.0504090.9664910.9160820.0335090.083918

70.0216040.9880950.9664910.0119050.033509

80.0081020.9961970.9880950.0038030.011905

90.0027010.9988980.9961970.0011020.003803

100.0008100.9997080.9988980.0002920.001102

110.0002210.9999290.9997080.0000710.000292

120.0000550.9999840.9999290.0000160.000071

130.0000130.9999970.9999840.0000030.000016

140.0000030.9999990.9999970.0000010.000003

150.0000011.0000000.9999990.0000000.000001

160.0000001.0000001.0000000.0000000.000000

170.0000001.0000001.0000000.0000000.000000

180.0000001.0000001.0000000.0000000.000000

190.0000001.0000001.0000000.0000000.000000

200.0000001.0000001.0000000.0000000.000000

&A

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Poisson2

0

0

0

0

0

0

0

0

x

P(x)

Poisson

0

0

0

0

0

0

0

0

0

0

0

0

x

P(x)

Sheet1

Poisson Probabilities for Customer Arrivals

Data

Average/Expected number of successes:0.1

Poisson Probabilities Table

XP(X)P(=X)

00.90480.9048370.0000000.0951631.000000

10.09050.9953210.9048370.0046790.095163

20.00450.9998450.9953210.0001550.004679

30.00020.9999960.9998450.0000040.000155

40.00001.0000000.9999960.0000000.000004

50.00001.0000001.0000000.0000000.000000

60.00001.0000001.0000000.0000000.000000

70.00001.0000001.0000000.0000000.000000

&A

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Sheet2

Sheet3

Chart2

0.6065306597

0.3032653299

0.0758163325

0.0126360554

0.0015795069

0.0001579507

0.0000131626

0.0000009402

x

P(x)

Histogram

0

0.6065306597

0.3032653299

0.0758163325

0.0126360554

0.0015795069

0.0001579507

0.0000131626

0.0000009402

0.0000000588

0.0000000033

0.0000000002

0

0

0

0

0

0

0

0

0

0

Number of Successes

P(X)

Histogram

Poisson2

Poisson Probabilities for Customer Arrivals

Data

Average/Expected number of successes:0.5

Poisson Probabilities Table

XP(X)P(=X)

00.6065310.6065310.0000000.3934691.000000

10.3032650.9097960.6065310.0902040.393469

20.0758160.9856120.9097960.0143880.090204

30.0126360.9982480.9856120.0017520.014388

40.0015800.9998280.9982480.0001720.001752

50.0001580.9999860.9998280.0000140.000172

60.0000130.9999990.9999860.0000010.000014

70.0000011.0000000.9999990.0000000.000001

80.0000001.0000001.0000000.0000000.000000

90.0000001.0000001.0000000.0000000.000000

100.0000001.0000001.0000000.0000000.000000

110.0000001.0000001.0000000.0000000.000000

120.0000001.0000001.0000000.0000000.000000

130.0000001.0000001.0000000.0000000.000000

140.0000001.0000001.0000000.0000000.000000

150.0000001.0000001.0000000.0000000.000000

160.0000001.0000001.0000000.0000000.000000

170.0000001.0000001.0000000.0000000.000000

180.0000001.0000001.0000000.0000000.000000

190.0000001.0000001.0000000.0000000.000000

200.0000001.0000001.0000000.0000000.000000

&A

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Poisson2

0

0

0

0

0

0

0

0

x

P(x)

Poisson

Poisson Probabilities for Customer Arrivals

Data

Average/Expected number of successes:0.1

Poisson Probabilities Table

XP(X)P(=X)

00.90480.9048370.0000000.0951631.000000

10.09050.9953210.9048370.0046790.095163

20.00450.9998450.9953210.0001550.004679

30.00020.9999960.9998450.0000040.000155

40.00001.0000000.9999960.0000000.000004

50.00001.0000001.0000000.0000000.000000

60.00001.0000001.0000000.0000000.000000

70.00001.0000001.0000000.0000000.000000

&A

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Sheet1

Sheet2

Sheet3

nh l PoissonCho X ~ B(n,p)

Dng phn phi Poisson xp x phn phi nh thc khi n >> p.

M hnh PoissonM hnh Poisson :+ Xt n php th Bernoulli. + Trong xc sut thnh cng l p.+ Cc php th c lp vi nhau.(Kt qu ca php th ny khng nh hng n kt qu ca cc php th kia)+ X s ln xut hin thnh cng trong n phpth.+ Trong n ln ( n 100) v p nh (p 0,01v np 20). Khi X ~ P(). Vi =np

M hnh PoissonV dTrong mt t tim chng cho 2000 tr em mt khu vc. Bit xc sut 1 tr b phn ng vi thuc khi tim l 0.001. Tnh xc sut trong 2000 tr c khng qu 1 tr b phn ng khi tim thuc.

Phn phi uTt c cc kh nng c th xy ra ca bin ngu nhin c phn phi u c xc sut bng nhau.X c phn phi u trong khong [a,b], k hiu X ~ U([a,b]).xminxmaxxf(x)Tng din tch min gii hn bi phn phi u l 1.0

Phn phi uHm mt xc sut ca phn phi u trong on [a,b]f(x) =vif(x) = gi tr hm mt ti im xa = gi tr nh nht ca xb = gi tr ln nht ca x

Phn phi uK vng

Phng sai

Phn phi uV d: Phn phi u trn khong 2 x 626.25f(x) = = .25 for 2 x 66 - 21xf(x)

Phn phi mBin ngu nhin T (t>0) gi l c phn phi m nu c hm mt xc sut

Vi s bin c xy ra trung bnh trong mt n v thi gian.t s n v thi gian cho n bin c k tip.e = 2.71828K hiu: T ~ exp(), T l khong thi gian gia 2 ln xy ra cc bin c.

Phn phi mHm phn phi xc sut

K vng v phng sai

Phn phi mV d: S khc hng n mt quy dch v vi t l l 15 ngi mt gi. Hi xc sut thi gian gia 2 khch hng lin tip n quy dch v t hn 3 pht l bao nhiu.Trung bnh c 15 khch hng n trong 1 gi, do = 153 pht = 0.05 giT: thi gian gia 2 khch hng lin tip n quy.P(T < .05) = 1 e- t = 1 e-(15)(.05) = 0.5276Vy c khong 52,76% khong thi gian gia 2 khch hng lin tip n lm dch v ti quy t hn 3 pht.

Phn phi mV d: Trong mt nh my sn xut linh kin in t, bit tui th ca mt mch in l bin ngu nhin c phn phi m vi tui th trung bnh l 6,25 nm. Nu thi gian bo hnh ca sn phm l 5 nm. Hi c bao nhiu % mch in ca nh my khi bn ra th trng phi thay th trc thi gian bo hnh.

Phn phi chunBin ngu nhin X nhn gi tr trong R gi l c phn phi chun vi tham s v 2 nu hm mt xc sut

Vi: EX = v VarX = 2.K hiu: X ~ N(, 2)

Phn phi chun Dng nh mt ci chung C tnh i xng Trung bnh = Trung v = Mode V tr ca phn phi c xc nh bi k vng, phn tn c xc nh bi lch tiu chun, Xc nh t + to Trung bnh = Trung v = Mode

Phn phi chunBng vic thay i cc tham s v , ta nhn c nhiu dng phn phi chun khc nhau

Chart3

00

0.002215924200.0000014867

0.002571320500.000002439

0.00297626620.00000000010.0000039613

0.00343638330.00000000010.0000063698

0.00395772580.00000000040.0000101409

0.00454678130.00000000090.0000159837

0.00521046740.00000000220.0000249425

0.00595612180.00000000530.0000385352

0.00679148460.00000001230.0000589431

0.00772467360.00000002820.0000892617

0.00876415020.00000006350.0001338302

0.00991867720.00000013990.0001986555

0.01119726510.00000030210.0002919469

0.012609110.00000063960.0004247803

0.01416351890.00000132730.0006119019

0.01586982590.00000269960.0008726827

0.01773729640.00000538230.0012322192

0.01977502080.00001051830.0017225689

0.0219917980.00002014830.0023840882

0.02439600930.00003783070.0032668191

0.02699548330.00006962490.0044318484

0.0297973530.00012560260.0059525324

0.03280790740.0002220990.0079154516

0.03603243720.00038495410.0104209348

0.03947507920.00065401160.0135829692

0.04313865940.00108912050.0175283005

0.04702453870.00177779050.0223945303

0.05113246230.00284445680.0283270377

0.05546041730.00446099980.0354745928

0.06000450030.00685771090.043983596

0.06475879780.01033332890.0539909665

0.06971528320.01526214050.0656158148

0.07486373280.02209554660.0789501583

0.08019166370.03135509790.0940490774

0.0856842960.04361397650.1109208347

0.09132454270.05946443790.1295175957

0.09709302750.07946997240.1497274656

0.10296813440.10410292120.171368592

0.10892608850.13367090270.194186055

0.11494107030.16823833270.217852177

0.12098536230.20755210430.2419707245

0.12702952820.25098249750.2660852499

0.13304262490.29749100760.2896915528

0.13899244310.34563551340.3122539334

0.14484577640.39361984480.3332246029

0.15056871610.43938954890.3520653268

0.15612696670.48076912580.3682701403

0.16148617980.51562922930.3813878155

0.16661230140.54206653460.391042694

0.17147192750.55857536080.3969525475

0.17603266340.56418958350.3989422804

0.18026348120.55857624590.3969525475

0.18413507020.54206825250.391042694

0.18762017350.51563168030.3813878155

0.19069390770.4807721730.3682701403

0.19333405840.439393030.3520653268

0.1955213470.3936235870.3332246029

0.19723966550.3456393470.3122539334

0.19847627370.29749477870.2896915528

0.1992219570.25098607670.2660852499

0.19947114020.20755539310.2419707245

0.1992219570.16824126510.217852177

0.19847627370.13367344430.194186055

0.19723966550.10410506560.171368592

0.1955213470.07947173530.1497274656

0.19333405840.05946585130.1295175957

0.19069390770.04361508220.1109208347

0.18762017350.03135594260.0940490774

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0.17603266340.01033365640.0539909665

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0.16661230140.00446115530.0354745928

0.16148617980.00284456050.0283270377

0.15612696670.00177785810.0223945303

0.15056871610.00108916370.0175283005

0.14484577640.00065403850.0135829692

0.13899244310.00038497060.0104209348

0.13304262490.00022210890.0079154516

0.12702952820.00012560830.0059525324

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0.10296813440.00001051890.0017225689

0.09709302750.00000538260.0012322192

0.09132454270.00000269980.0008726827

0.0856842960.00000132730.0006119019

0.08019166370.00000063970.0004247803

0.07486373280.00000030220.0002919469

0.06971528320.00000013990.0001986555

0.06475879780.00000006350.0001338302

0.06000450030.00000002820.0000892617

0.05546041730.00000001230.0000589431

0.05113246230.00000000530.0000385352

0.04702453870.00000000220.0000249425

0.04313865940.00000000090.0000159837

0.03947507920.00000000040.0000101409

0.03603243720.00000000010.0000063698

0.03280790740.00000000010.0000039613

0.02979735300.000002439

0.026995483300.0000014867

0.024396009300.0000008972

Sheet1

xpopulationsampling populationsigman

-50.002215924200.0000014867281

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-4.60.00395772580.00000000040.0000101409

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1.60.19069390770.04361508220.1109208347

1.70.18762017350.03135594260.0940490774

1.80.18413507020.02209617680.0789501583

1.90.18026348120.01526260.0656158148

20.17603266340.01033365640.0539909665

2.10.17147192750.00685793910.043983596

2.20.16661230140.00446115530.0354745928

2.30.16148617980.00284456050.0283270377

2.40.15612696670.00177785810.0223945303

2.50.15056871610.00108916370.0175283005

2.60.14484577640.00065403850.0135829692

2.70.13899244310.00038497060.0104209348

2.80.13304262490.00022210890.0079154516

2.90.12702952820.00012560830.0059525324

30.12098536230.00006962820.0044318484

3.10.11494107030.00003783260.0032668191

3.20.10892608850.00002014930.0023840882

3.30.10296813440.00001051890.0017225689

3.40.09709302750.00000538260.0012322192

3.50.09132454270.00000269980.0008726827

3.60.0856842960.00000132730.0006119019

3.70.08019166370.00000063970.0004247803

3.80.07486373280.00000030220.0002919469

3.90.06971528320.00000013990.0001986555

40.06475879780.00000006350.0001338302

4.10.06000450030.00000002820.0000892617

4.20.05546041730.00000001230.0000589431

4.30.05113246230.00000000530.0000385352

4.40.04702453870.00000000220.0000249425

4.50.04313865940.00000000090.0000159837

4.60.03947507920.00000000040.0000101409

4.70.03603243720.00000000010.0000063698

4.80.03280790740.00000000010.0000039613

4.90.02979735300.000002439

50.026995483300.0000014867

5.10.024396009300.0000008972

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Phn phi chunxf(x)Thay i dch chuyn phn phi qua tri hoc phiThay i lm tng hoc gim phn tn.

Hm phn phi ca phn phi chunXt bin ngu nhin X c phn phi chun vi trung bnh v phng sai 2 , X~N(, 2), hm phn phi ca X l

f(x)

Xc sut ca phn phi chunXc sut X (a,b) o bi din tch gii hn bi ng cong chun.ba

Xc sut ca phn phi chun

Phn phi chun haXt bin ngu nhin X ~ N(, 2). Chun ha X bng cch t

Khi EZ = 0 v VarZ = 1. Ta ni Z c phn phi chun ha. K hiu

Phn phi chun haNu X c phn phi chun vi trung bnh l 100 and lch tiu chun l 50, th gi tr ca Z ng vi X = 200 l

Z1002.00200X( = 100, = 50)( = 0, = 1)

Phn phi chun haHm mt

Hm phn phi

Tnh xc sut

Tnh xc sut

Tra bng chun ha N(0,1) tm xc xut P(X

Tra bng chun ha N(0,1)P(Z

Tra bng chun ha N(0,1)V d: P(Z < 2.00) = (2.00) = .9772Z02.00.9772Do tnh i xng(-z) = 1 - (z) V d: P(Z < -2.00) = (-2.00)= 1 (2.00) = 1 - 0.9772 = 0.0228

V dGi s X c phn phi chun vi trung bnh l 8.0 v lch tiu chun 5.0. Tm P(X < 8.6).X8.68.0

V dZ0.12 0X8.6 8 = 8 = 10 = 0 = 1P(X < 8.6)P(Z < 0.12)

V dZ0.12z(z).10.5398.11.5438.12.5478.13.5517(0.12) = 0.5478Tra bng chun ha0.00= P(Z < 0.12)P(X < 8.6)

V dGi s X c phn phi chun vi trung bnh 8.0 v lch tiu chun 5.0. Tm P(X > 8.6)

V dTm P(X > 8.6)Z0.12 0Z0.5478 01.0001.0 - 0.5478 = 0.4522 P(X > 8.6) = P(Z > 0.12) = 1.0 - P(Z 0.12) = 1.0 - 0.5478 = 0.45220.12

Quy tc k -

Xp x phn phi nh thc bng phn phi chunCho X ~ B(n,p). Khi n ln v p khng qu gn 0 v 1.Tnh P(X < c)?Tnh P(a < X < b)?Dng phn phi chun.

Xp x phn phi nh thc bng phn phi chunt = EX = np2 = VarX = np(1-p)To bin ngu nhin Z c phn phi chun ha t phn phi nh thc

Xp x phn phi nh thc bng phn phi chun

Xp x phn phi nh thc bng phn phi chunV dTrong mt cuc bu c mt thnh ph, bit rng 40% ngi dn ng h ng c vin A. Chn ngu nhin 200 ngi, hi xc sut gp c t 76 n 80 ngi ng h ng c vin A l bao nhiu?

V d

E(X) = = nP = 200(0.40) = 80Var(X) = 2 = nP(1 P) = 200(0.40)(1 0.40) = 48