Métodos Estatísticos II - Variáveis Aleatórias

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  • Mtodos Estatsticos II

    Prof.: Manuel Rosa de Oliveira Lino

    Variveis Aleatrias

    Resumo Unidade 3

  • Na realizao de um fenmeno aleatrio, comum ter-se interesse em uma ou mais quantidades, estas so funes de ocorrncias do fenmeno.

    Antes da realizao do fenmeno, no se sabe seu resultado, entretanto seu espao de probabilidade pode ser estabelecido de forma conveniente.

    Uma varivel aleatria uma funo do espao amostral S nos reais, para a qual possvel calcular a probabilidade da ocorrncia de seus valores.Definio

  • Uma varivel aleatria pode ser explicada por uma funo que associa a cada elemento sS um nmero real X(s).Conceito

  • Uma varivel aleatria pode ser dita discreta (v.a.d.) quando puder assumir valores num conjunto enumervel de pontos.

    Variveis aleatrias discretas (v.a.d.) contagens.

    Peas defeituosas em um lote;Chegadas a um posto de informaes tursticas;Produtos rejeitados em um teste de qualidade;Erros em um livro;Defeitos em bobinas de alumnio;Aprovaes em um teste psicotcnico;Pontos obtidos, na soma, no lanamento de dois dados.Varivel aleatria discreta:

  • Considere o experimento : lanamento de duas moedas e observao das faces voltadas para cima, cujo espao amostral S, ser dado porS={(C,C), (C,K), (K,C), (K,K)}.

    Considere a observao de X, X: nmero de caras obtido nos lanamentos, cujos resultados possveis sero dados por X={0, 1, 2} Exemplo 1:

  • Considere o experimento : sortear trs alunos de uma turma de um plo de EAD constitudo por 10 homens e 20 mulheres e o registro do sexo dos sorteados.

    Considere a observao da v.a. X, X: nmero de homens obtido nos sorteios. Exemplo 2:

  • esquerda foi construda uma rvore de possibilidades.

    O espao amostral a partir da rvore de possibilidades ser dado por:S={(HHH),(HHM),(HMH),(HMM),(MHH),(MHM),(MMH),(MMM)}.

    Assim a varivel aleatria X, poder assumir os seguintes valores:X={0, 1, 2, 3}Exemplo 2:

  • Considere a seguinte situao:Uma pea montada a partir de dois componentes produzidos por diferentes indstrias, os quais podem ser classificados segundo um critrio de qualidade.Exemplo 3:Os componentes sua origem, classificao e distribuio esto apresentados ao lado.

    IndstriaQualidadeComp.Ind. AComp. Ind. BBomReaproveitvelDescartvel0,800,150,050,700,200,10

  • Este modelo caracteriza uma realidade e est baseado em suposies. A distribuio encontrada, uma distribuio terica, explicar to bem a distribuio de frequncias real quanto mais prximas da realidade forem as suposies.

    O espao amostral (pares ordenados) esto construdos a abaixo partir da qualidade de cada um dos componentes e as probabilidades associadas a cada par podero ser atribudas. Exemplo 3:

    RelaoBBBRBDRBRRRDDBDRDD

  • Uma varivel aleatria pode ser dita continua (v.a.c.) quando puder assumir valores num conjunto infinito de pontos.

    Variveis aleatrias continuas medies.

    Altura, peso (idade ?) de um estudante;Quilometragem de um pneu at o limite permitido por lei;Tempo de anlise de um balancete;Peso de correspondncias enviadas por Sedex;Volume de precipitao pluviomtrica num dia especificado;rea necessria para alocao de 500 pessoas sentadas;Valor despendido em uma programao festiva. Varivel aleatria contnua:

  • Considere a seguinte situao:Observar o tempo de durao em minutos de espera em uma fila de um banco.Exemplo 4:A funo distribuio da durao sendo dada por:

    Assim a varivel aleatria X, poder assumir os seguintes valores: X={x|xR, 0x50} (Gilberto de Andrade Martins Atlas)

  • Considere a seguinte situao:Observar o tempo de durao em anos de uma lmpada para iluminao pblica.Exemplo 5:A funo distribuio da durao sendo dada por:

    Assim a varivel aleatria X, poder assumir os seguintes valores: X={x|xR, x0} (Marcos Antonio Magalhes - Edusp)

  • Considere a seguinte situao:Observar o peso de bagagens, em kg, num voo de uma de companhia em Florianpolis em 13/04/09. Exemplo 6:A funo distribuio da durao sendo dada por:

    Assim a varivel aleatria X, poder assumir os seguintes valores: X={x|xR, 10kgx50kg} (=22,5 e =4,25)

  • A v.a. X, do Ex. 1, possui distribuio dada por:Distribuio de probabilidade de xDistribuio de probabilidade V.A.D.

    X012Faces CCCK KCKKP(X=xi)0,250,500,25

    Grf2

    0.25

    0.5

    0.25

    p(X)

    X

    P(X=x )

    Distribuio de probabilidade de X

    Plan1

    20.115.716.918.218.517.415.219.417.916.416.817.518.6

    17.61819.618.220.521.417.617.518.318.518.919.418.8

    18.618.916.518.519.318.619.718.818.518.719.614.522.7

    19.320.119.217.220.618.618.519.218.518.917.719.9

    1234567891011121314151617181920

    1SSSSSSSSSSSSSSNSSSSS

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    10SNSNSSSNSSSNSSNSNSSS

    11NSSSSSSNSSSSSSNSSSSS

    12SNSSSNSSNSSSNSSSNSSS

    13SSNSNNSSNS

    123456789101112131415161718192021

    1BBBBBBBBBBBBBBBBBBBBB

    2BBBBBBBBBBBBBBBBBBBBB

    3BBBBBBBBBBDBBBBBBBBBB

    ............

    79BBBBBBBBBBBBBBBBBBBBB

    80BBBBBBBBBBBBBBBBBBBBB

    6.15.84.97.586.45.85.26.2

    5.95.76.35.65.95.84.46

    Moedas

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    9790.11940.82481.0

    9800.90420.34820.0

    9810.47230.73751.0

    9820.30470.35250.0

    9830.27430.81801.0

    9840.58570.21600.0

    9850.19000.21880.0

    9860.46560.21320.0

    9870.19480.53421.0

    9880.07950.77901.0

    9890.20430.60991.0

    9900.64470.73081.0

    9910.18280.71631.0

    9920.99190.83821.0

    9930.49360.63261.0

    9940.54470.43280.0

    9950.61960.30250.0

    9960.32640.96261.0

    9970.07420.20360.0

    9980.80800.40320.0

    9990.87980.72821.0

    10000.31410.43470.0

    Moedas

    1

    0.5

    0.3333333333

    0.25

    0.4

    0.3333333333

    0.2857142857

    0.25

    0.3333333333

    0.4

    0.3636363636

    0.4166666667

    0.3846153846

    0.4285714286

    0.4

    0.375

    0.4117647059

    0.3888888889

    0.4210526316

    0.4

    0.4285714286

    0.4090909091

    0.3913043478

    0.4166666667

    0.44

    0.4615384615

    0.4814814815

    0.5

    0.4827586207

    0.5

    0.4838709677

    0.5

    0.4848484848

    0.4705882353

    0.4857142857

    0.4722222222

    0.4864864865

    0.4736842105

    0.4871794872

    0.475

    0.4634146341

    0.4523809524

    0.4418604651

    0.4318181818

    0.4444444444

    0.4565217391

    0.4468085106

    0.4375

    0.4285714286

    0.44

    0.431372549

    0.4423076923

    0.4339622642

    0.4444444444

    0.4363636364

    0.4464285714

    0.4385964912

    0.4482758621

    0.4576271186

    0.45

    0.4590163934

    0.4516129032

    0.4444444444

    0.4375

    0.4307692308

    0.4242424242

    0.4328358209

    0.4411764706

    0.4347826087

    0.4285714286

    0.4225352113

    0.4305555556

    0.4246575342

    0.4324324324

    0.44

    0.4473684211

    0.4415584416

    0.4358974359

    0.4303797468

    0.425

    0.4197530864

    0.4146341463

    0.421686747

    0.4285714286

    0.4352941176

    0.4302325581

    0.4252873563

    0.4318181818

    0.4382022472

    0.4333333333

    0.4285714286

    0.4347826087

    0.4301075269

    0.4361702128

    0.4315789474

    0.4270833333

    0.4226804124

    0.4183673469

    0.4242424242

    0.43

    0.4257425743

    0.4215686275

    0.427184466

    0.4326923077

    0.4285714286

    0.4245283019

    0.4299065421

    0.4351851852

    0.4403669725

    0.4363636364

    0.4414414414

    0.4464285714

    0.4513274336

    0.4473684211

    0.4434782609

    0.4396551724

    0.4444444444

    0.4491525424

    0.4537815126

    0.45

    0.4462809917

    0.4426229508

    0.4471544715

    0.4516129032

    0.448

    0.4523809524

    0.4566929134

    0.453125

    0.4496124031

    0.4461538462

    0.4427480916

    0.4393939394

    0.4360902256

    0.4328358209

    0.4296296296

    0.4264705882

    0.4306569343

    0.4347826087

    0.4316546763

    0.4357142857

    0.4397163121

    0.4436619718

    0.4475524476

    0.4513888889

    0.4551724138

    0.4520547945

    0.4489795918

    0.4527027027

    0.4496644295

    0.4466666667

    0.4437086093

    0.4407894737

    0.4379084967

    0.4415584416

    0.4451612903

    0.4423076923

    0.4458598726

    0.4430379747

    0.4465408805

    0.44375

    0.4409937888

    0.4444444444

    0.4417177914

    0.4390243902

    0.4363636364

    0.4337349398

    0.4371257485

    0.4404761905

    0.4378698225

    0.4352941176

    0.4385964912

    0.4360465116

    0.4393063584

    0.4425287356

    0.4457142857

    0.4488636364

    0.4463276836

    0.4438202247

    0.4469273743

    0.4444444444

    0.4475138122

    0.4450549451

    0.4426229508

    0.4456521739

    0.4432432432

    0.4462365591

    0.4438502674

    0.4414893617

    0.4444444444

    0.4473684211

    0.4502617801

    0.453125

    0.4507772021

    0.4484536082

    0.4461538462

    0.4489795918

    0.4467005076

    0.4494949495

    0.4522613065

    0.45

    0.447761194

    0.4504950495

    0.4532019704

    0.4558823529

    0.4536585366

    0.4514563107

    0.4541062802

    0.4519230769

    0.4545454545

    0.4523809524

    0.4549763033

    0.4575471698

    0.4600938967

    0.4626168224

    0.4651162791

    0.4675925926

    0.465437788

    0.4678899083

    0.4657534247

    0.4681818182

    0.4705882353

    0.4684684685

    0.466367713

    0.4642857143

    0.4622222222

    0.4646017699

    0.4669603524

    0.4649122807

    0.4672489083

    0.4695652174

    0.4675324675

    0.4698275862

    0.4721030043

    0.4700854701

    0.4723404255

    0.4703389831

    0.4725738397

    0.474789916

    0.4769874477

    0.475

    0.4771784232

    0.4752066116

    0.4773662551

    0.4754098361

    0.4775510204

    0.4796747967

    0.4817813765

    0.4838709677

    0.4819277108

    0.484

    0.4860557769

    0.4880952381

    0.4861660079

    0.4881889764

    0.4862745098

    0.484375

    0.4824902724

    0.480620155

    0.4826254826

    0.4846153846

    0.4827586207

    0.4809160305

    0.4790874525

    0.4810606061

    0.4830188679

    0.4812030075

    0.4794007491

    0.4776119403

    0.4758364312

    0.4740740741

    0.4760147601

    0.4779411765

    0.4761904762

    0.4744525547

    0.4727272727

    0.4710144928

    0.4693140794

    0.4712230216

    0.4731182796

    0.475

    0.4733096085

    0.475177305

    0.4770318021

    0.4788732394

    0.4807017544

    0.4825174825

    0.4808362369

    0.4791666667

    0.4775086505

    0.4793103448

    0.4776632302

    0.4794520548

    0.4778156997

    0.4795918367

    0.4813559322

    0.4831081081

    0.4814814815

    0.4798657718

    0.4816053512

    0.4833333333

    Lanamentos

    Freq. relativa

    Dist acum

    000

    10.166670.16667

    20.166670.33333

    30.166670.50000

    40.166670.66667

    50.166670.83333

    60.166671.00000

    6

    Dist acum

    0

    0.1666666667

    0.3333333333

    0.5

    0.6666666667

    0.8333333333

    1

    Plan4

    Plan2

    Plan3

    Xp(X)

    00.25

    10.5

    20.25

    Plan3

    0.25

    0.5

    0.25

    p(X)

    X

    P(X=x )

    Distribuio de probabilidade de X

  • A v.a. X, do Ex. 2, possui distribuio dada por:

    P(X=0)=(2/3)3, P(X=1)=(2/3)2(1/3)13,P(X=2)=(2/3)1(1/3)23 e P(X=3)=(1/3)3Distribuio de probabilidade V.A.D.(ou atravs do grfico faa-o):

    X0123P(X=xi)0,29630,44440,22220,0370

  • Dada a v.a.d. X, de valores x1, x2, , xk, com as respectivas probabilidades p1, p2, ..., pk chama-se:Esperana e Desvio padro da V.A.D.Esperana:

    Varincia:

    Desvio padro:

  • Esperana e Desvio padro da V.A.D.A v.a. X, do Ex. 1 possui E(X) e V(X) dados por:

    X012SomaP(X=xi)0,250,500,251,00xi * P(xi)0,000,500,501,00x2i * P(xi)0,000,502,002,50

  • Esperana e Desvio padro da V.A.D.A v.a. X, do Ex. 3 possui E(X) e V(X) dados por:

  • Taxa de retorno esperada ???A distribuio de probabilidade abaixo representa o retorno de um investimento baseado no resultado da economia.Distribuio de probabilidade V.A.D.Valor esperado e Varincia

    Resultado EconmicoPreviso Probab.Retorno FinanceiroPond.BomRazovelRuimMuito ruim0,200,400,300,1025%15%05%00%5,0%6,0%1,5%0,0%

  • Observe que no caso de uma v.a.c. qualquer X, no possvel listar todos os valores por serem infinitos.Funo de dist. de prob. V.A.C.A funo distribuio pode ser definida como a probabilidade de X ser menor ou igual a xi, ou seja: A probabilidade de X estar compreendido no intervalo dado pelos pontos a e b ser dada por:

  • A v.a. X, do Ex. 4, possui distribuio dada pelo grfico:Funo de dist. de prob. V.A.C.

    Grf1

    0

    0.0951604666

    0.1812654188

    0.2591765835

    0.3296736855

    0.3934622504

    0.4511806656

    0.5034065696

    0.5506626321

    0.5934217857

    0.6321119583

    0.667120356

    0.6987973382

    0.7274599241

    0.7533949649

    0.7768620151

    0.7980959298

    0.8173092153

    0.8346941557

    0.8504247369

    0.8646583888

    0.8775375596

    0.8891911426

    0.8997357652

    0.9092769565

    0.9179102037

    0.925721907

    0.932790245

    0.9391859566

    0.9449730494

    0.9502094397

    0.9549475326

    0.9592347464

    0.963113987

    0.9666240772

    0.9698001456

    0.9726739778

    0.9752743348

    0.9776272407

    0.9797562429

    0.9816826483

    0.983425736

    0.9850029507

    0.9864300769

    0.9877213971

    0.9888898347

    0.9899470832

    0.9909037235

    0.9917693294

    0.9925525638

    0.9932612653

    0.9939025265

    0.9944827649

    0.9950077876

    0.9954828488

    0.995912703

    0.9963016521

    0.9966535886

    0.9969720347

    0.9972601773

    0.9975209001

    0.9977568124

    0.9979702752

    0.9981634247

    0.9983381941

    0.9984963323

    0.998639422

    0.9987688953

    0.9988860478

    0.998992052

    0.9990879688

    Plan1

    xf(x)xf(x)

    -1-0.2210.10.0198

    -0.5-0.1050.20.0392

    000.30.0582

    0.50.09520.40.0769

    10.18130.50.0952

    1.50.25920.60.1131

    20.32970.70.1306

    2.50.39350.80.1479

    30.45120.90.1647

    3.50.503410.1813

    40.55071.10.1975

    4.50.59341.20.2134

    50.63211.30.2289

    5.50.66711.40.2442

    60.69881.50.2592

    6.50.72751.60.2739

    70.75341.70.2882

    7.50.77691.80.3023

    80.79811.90.3161

    8.50.817320.3297

    90.83472.10.3430

    9.50.85042.20.3560

    100.86472.30.3687

    10.50.87752.40.3812

    110.88922.50.3935

    11.50.89972.60.4055

    120.90932.70.4173

    12.50.91792.80.4288

    130.92572.90.4401

    13.50.932830.45120.3469

    140.93923.10.4621

    14.50.94503.20.4727

    150.95023.30.4831

    15.50.95493.40.4934

    160.95923.50.5034

    16.50.96313.60.5132

    170.96663.70.5229

    17.50.96983.80.5323

    180.97273.90.5416

    18.50.975340.5507

    190.97764.10.5596

    19.50.97984.20.5683

    200.98174.30.5768

    20.50.98344.40.5852

    210.98504.50.5934

    21.50.98644.60.6015

    220.98774.70.6094

    22.50.98894.80.6171

    230.98994.90.6247

    23.50.990950.6321

    240.99185.10.6394

    24.50.99265.20.6465

    250.99335.30.6535

    25.50.99395.40.6604

    260.99455.50.6671

    26.50.99505.60.6737

    270.99555.70.6802

    27.50.99595.80.6865

    280.99635.90.6927

    28.50.996760.6988

    290.99706.10.7048

    29.50.99736.20.7106

    300.99756.30.7163

    30.50.99786.40.7220

    310.99806.50.7275

    31.50.99826.60.7329

    320.99836.70.7382

    32.50.99856.80.7433

    330.99866.90.7484

    33.50.998870.7534

    340.99897.10.7583

    34.50.99907.20.7631

    350.99917.30.7678

    35.50.99927.40.7724

    360.99937.50.7769

    36.50.99937.60.7813

    370.99947.70.7856

    37.50.99947.80.7899

    380.99957.90.7940

    38.50.999580.7981

    390.99968.10.8021

    39.50.99968.20.8060

    400.99978.30.8099

    40.50.99978.40.8136

    410.99978.50.8173

    41.50.99988.60.8209

    420.99988.70.8245

    42.50.99988.80.8280

    430.99988.90.8314

    43.50.999890.8347

    440.99989.10.8380

    44.50.99999.20.8412

    450.99999.30.8443

    45.50.99999.40.8474

    460.99999.50.8504

    46.50.99999.60.8534

    470.99999.70.8563

    47.50.99999.80.8591

    480.99999.90.8619

    48.50.9999100.8647

    490.999910.10.8673

    49.50.999910.20.8700

    501.000010.30.8725

    10.40.8751

    10.50.8775

    10.60.8800

    10.70.8823

    10.80.8847

    10.90.8870

    110.8892

    11.10.8914

    11.20.8935

    11.30.8956

    11.40.8977

    11.50.8997

    11.60.9017

    11.70.9037

    11.80.9056

    11.90.9074

    120.9093

    12.10.9111

    12.20.9128

    12.30.9146

    12.40.9163

    12.50.9179

    12.60.9195

    12.70.9211

    12.80.9227

    12.90.9242

    130.9257

    13.10.9272

    13.20.9286

    13.30.9301

    13.40.9314

    13.50.9328

    13.60.9341

    13.70.9354

    13.80.9367

    13.90.9380

    140.9392

    14.10.9404

    14.20.9416

    14.30.9427

    14.40.9439

    14.50.9450

    14.60.9461

    14.70.9471

    14.80.9482

    14.90.9492

    150.9502

    15.10.9512

    15.20.9522

    15.30.9531

    15.40.9540

    15.50.9550

    15.60.9558

    15.70.9567

    15.80.9576

    15.90.9584

    160.9592

    16.10.9600

    16.20.9608

    16.30.9616

    16.40.9624

    16.50.9631

    16.60.9638

    16.70.9646

    16.80.9653

    16.90.9660

    170.9666

    17.10.9673

    17.20.9679

    17.30.9686

    17.40.9692

    17.50.9698

    17.60.9704

    17.70.9710

    17.80.9716

    17.90.9721

    180.9727

    18.10.9732

    18.20.9737

    18.30.9743

    18.40.9748

    18.50.9753

    18.60.9758

    18.70.9762

    18.80.9767

    18.90.9772

    190.9776

    19.10.9781

    19.20.9785

    19.30.9789

    19.40.9793

    19.50.9798

    19.60.9802

    19.70.9806

    19.80.9809

    19.90.9813

    200.9817

    20.10.9820

    20.20.9824

    20.30.9828

    20.40.9831

    20.50.9834

    20.60.9838

    20.70.9841

    20.80.9844

    20.90.9847

    210.9850

    21.10.9853

    21.20.9856

    21.30.9859

    21.40.9862

    21.50.9864

    21.60.9867

    21.70.9870

    21.80.9872

    21.90.9875

    220.9877

    22.10.9880

    22.20.9882

    22.30.9884

    22.40.9887

    22.50.9889

    22.60.9891

    22.70.9893

    22.80.9895

    22.90.9897

    230.9899

    23.10.9901

    23.20.9903

    23.30.9905

    23.40.9907

    23.50.9909

    23.60.9911

    23.70.9913

    23.80.9914

    23.90.9916

    240.9918

    24.10.9919

    24.20.9921

    24.30.9922

    24.40.9924

    24.50.9926

    24.60.9927

    24.70.9928

    24.80.9930

    24.90.9931

    250.9933

    25.10.9934

    25.20.9935

    25.30.9937

    25.40.9938

    25.50.9939

    25.60.9940

    25.70.9941

    25.80.9943

    25.90.9944

    260.9945

    26.10.9946

    26.20.9947

    26.30.9948

    26.40.9949

    26.50.9950

    26.60.9951

    26.70.9952

    26.80.9953

    26.90.9954

    270.9955

    27.10.9956

    27.20.9957

    27.30.9957

    27.40.9958

    27.50.9959

    27.60.9960

    27.70.9961

    27.80.9962

    27.90.9962

    280.9963

    28.10.9964

    28.20.9964

    28.30.9965

    28.40.9966

    28.50.9967

    28.60.9967

    28.70.9968

    28.80.9968

    28.90.9969

    290.9970

    29.10.9970

    29.20.9971

    29.30.9971

    29.40.9972

    29.50.9973

    29.60.9973

    29.70.9974

    29.80.9974

    29.90.9975

    300.9975

    30.10.9976

    30.20.9976

    30.30.9977

    30.40.9977

    30.50.9978

    30.60.9978

    30.70.9978

    30.80.9979

    30.90.9979

    310.9980

    31.10.9980

    31.20.9981

    31.30.9981

    31.40.9981

    31.50.9982

    31.60.9982

    31.70.9982

    31.80.9983

    31.90.9983

    320.9983

    32.10.9984

    32.20.9984

    32.30.9984

    32.40.9985

    32.50.9985

    32.60.9985

    32.70.9986

    32.80.9986

    32.90.9986

    330.9986

    33.10.9987

    33.20.9987

    33.30.9987

    33.40.9987

    33.50.9988

    33.60.9988

    33.70.9988

    33.80.9988

    33.90.9989

    340.9989

    34.10.9989

    34.20.9989

    34.30.9990

    34.40.9990

    34.50.9990

    34.60.9990

    34.70.9990

    34.80.9991

    34.90.9991

    350.9991

    35.10.9991

    35.20.9991

    35.30.9991

    35.40.9992

    35.50.9992

    35.60.9992

    35.70.9992

    35.80.9992

    35.90.9992

    360.9993

    36.10.9993

    36.20.9993

    36.30.9993

    36.40.9993

    36.50.9993

    36.60.9993

    36.70.9994

    36.80.9994

    36.90.9994

    370.9994

    37.10.9994

    37.20.9994

    37.30.9994

    37.40.9994

    37.50.9994

    37.60.9995

    37.70.9995

    37.80.9995

    37.90.9995

    380.9995

    38.10.9995

    38.20.9995

    38.30.9995

    38.40.9995

    38.50.9995

    38.60.9996

    38.70.9996

    38.80.9996

    38.90.9996

    390.9996

    39.10.9996

    39.20.9996

    39.30.9996

    39.40.9996

    39.50.9996

    39.60.9996

    39.70.9996

    39.80.9997

    39.90.9997

    400.9997

    40.10.9997

    40.20.9997

    40.30.9997

    40.40.9997

    40.50.9997

    40.60.9997

    40.70.9997

    40.80.9997

    40.90.9997

    410.9997

    41.10.9997

    41.20.9997

    41.30.9997

    41.40.9997

    41.50.9998

    41.60.9998

    41.70.9998

    41.80.9998

    41.90.9998

    420.9998

    42.10.9998

    42.20.9998

    42.30.9998

    42.40.9998

    42.50.9998

    42.60.9998

    42.70.9998

    42.80.9998

    42.90.9998

    430.9998

    43.10.9998

    43.20.9998

    43.30.9998

    43.40.9998

    43.50.9998

    43.60.9998

    43.70.9998

    43.80.9998

    43.90.9998

    440.9998

    44.10.9999

    44.20.9999

    44.30.9999

    44.40.9999

    44.50.9999

    44.60.9999

    44.70.9999

    44.80.9999

    44.90.9999

    450.9999

    45.10.9999

    45.20.9999

    45.30.9999

    45.40.9999

    45.50.9999

    45.60.9999

    45.70.9999

    45.80.9999

    45.90.9999

    460.9999

    46.10.9999

    46.20.9999

    46.30.9999

    46.40.9999

    46.50.9999

    46.60.9999

    46.70.9999

    46.80.9999

    46.90.9999

    470.9999

    47.10.9999

    47.20.9999

    47.30.9999

    47.40.9999

    47.50.9999

    47.60.9999

    47.70.9999

    47.80.9999

    47.90.9999

    480.9999

    48.10.9999

    48.20.9999

    48.30.9999

    48.40.9999

    48.50.9999

    48.60.9999

    48.70.9999

    48.80.9999

    48.90.9999

    490.9999

    49.10.9999

    49.20.9999

    49.30.9999

    49.40.9999

    49.50.9999

    49.61.0000

    49.71.0000

    Plan1

    Plan2

    Plan3

  • Distribuio de probabilidade V.A.C.A v.a. X, do Ex. 6, possui distribuio dada pelo grfico (observe que trata-se de uma distribuio de frequncias relativas simples):

    Grf2

    0

    0.0202020202

    0.0757575758

    0.1818181818

    0.2727272727

    0.2474747475

    0.1464646465

    0.0505050505

    0.0050505051

    0

    X=kg

    P(X)

    Exp 0,2

    xf(x)xf(x)

    -1-0.2210.10.0198

    -0.5-0.1050.20.0392

    000.30.0582

    0.50.09520.40.0769

    10.18130.50.0952

    1.50.25920.60.1131

    20.32970.70.1306

    2.50.39350.80.1479

    30.45120.90.1647

    3.50.503410.1813

    40.55071.10.1975

    4.50.59341.20.2134

    50.63211.30.2289

    5.50.66711.40.2442

    60.69881.50.2592

    6.50.72751.60.2739

    70.75341.70.2882

    7.50.77691.80.3023

    80.79811.90.3161

    8.50.817320.3297

    90.83472.10.3430

    9.50.85042.20.3560

    100.86472.30.3687

    10.50.87752.40.3812

    110.88922.50.3935

    11.50.89972.60.4055

    120.90932.70.4173

    12.50.91792.80.4288

    130.92572.90.4401

    13.50.932830.45120.3469

    140.93923.10.4621

    14.50.94503.20.4727

    150.95023.30.4831

    15.50.95493.40.4934

    160.95923.50.5034

    16.50.96313.60.5132

    170.96663.70.5229

    17.50.96983.80.5323

    180.97273.90.5416

    18.50.975340.5507

    190.97764.10.5596

    19.50.97984.20.5683

    200.98174.30.5768

    20.50.98344.40.5852

    210.98504.50.5934

    21.50.98644.60.6015

    220.98774.70.6094

    22.50.98894.80.6171

    230.98994.90.6247

    23.50.990950.6321

    240.99185.10.6394

    24.50.99265.20.6465

    250.99335.30.6535

    25.50.99395.40.6604

    260.99455.50.6671

    26.50.99505.60.6737

    270.99555.70.6802

    27.50.99595.80.6865

    280.99635.90.6927

    28.50.996760.6988

    290.99706.10.7048

    29.50.99736.20.7106

    300.99756.30.7163

    30.50.99786.40.7220

    310.99806.50.7275

    31.50.99826.60.7329

    320.99836.70.7382

    32.50.99856.80.7433

    330.99866.90.7484

    33.50.998870.7534

    340.99897.10.7583

    34.50.99907.20.7631

    350.99917.30.7678

    35.50.99927.40.7724

    360.99937.50.7769

    36.50.99937.60.7813

    370.99947.70.7856

    37.50.99947.80.7899

    380.99957.90.7940

    38.50.999580.7981

    390.99968.10.8021

    39.50.99968.20.8060

    400.99978.30.8099

    40.50.99978.40.8136

    410.99978.50.8173

    41.50.99988.60.8209

    420.99988.70.8245

    42.50.99988.80.8280

    430.99988.90.8314

    43.50.999890.8347

    440.99989.10.8380

    44.50.99999.20.8412

    450.99999.30.8443

    45.50.99999.40.8474

    460.99999.50.8504

    46.50.99999.60.8534

    470.99999.70.8563

    47.50.99999.80.8591

    480.99999.90.8619

    48.50.9999100.8647

    490.999910.10.8673

    49.50.999910.20.8700

    501.000010.30.8725

    10.40.8751

    10.50.8775

    10.60.8800

    10.70.8823

    10.80.8847

    10.90.8870

    110.8892

    11.10.8914

    11.20.8935

    11.30.8956

    11.40.8977

    11.50.8997

    11.60.9017

    11.70.9037

    11.80.9056

    11.90.9074

    120.9093

    12.10.9111

    12.20.9128

    12.30.9146

    12.40.9163

    12.50.9179

    12.60.9195

    12.70.9211

    12.80.9227

    12.90.9242

    130.9257

    13.10.9272

    13.20.9286

    13.30.9301

    13.40.9314

    13.50.9328

    13.60.9341

    13.70.9354

    13.80.9367

    13.90.9380

    140.9392

    14.10.9404

    14.20.9416

    14.30.9427

    14.40.9439

    14.50.9450

    14.60.9461

    14.70.9471

    14.80.9482

    14.90.9492

    150.9502

    15.10.9512

    15.20.9522

    15.30.9531

    15.40.9540

    15.50.9550

    15.60.9558

    15.70.9567

    15.80.9576

    15.90.9584

    160.9592

    16.10.9600

    16.20.9608

    16.30.9616

    16.40.9624

    16.50.9631

    16.60.9638

    16.70.9646

    16.80.9653

    16.90.9660

    170.9666

    17.10.9673

    17.20.9679

    17.30.9686

    17.40.9692

    17.50.9698

    17.60.9704

    17.70.9710

    17.80.9716

    17.90.9721

    180.9727

    18.10.9732

    18.20.9737

    18.30.9743

    18.40.9748

    18.50.9753

    18.60.9758

    18.70.9762

    18.80.9767

    18.90.9772

    190.9776

    19.10.9781

    19.20.9785

    19.30.9789

    19.40.9793

    19.50.9798

    19.60.9802

    19.70.9806

    19.80.9809

    19.90.9813

    200.9817

    20.10.9820

    20.20.9824

    20.30.9828

    20.40.9831

    20.50.9834

    20.60.9838

    20.70.9841

    20.80.9844

    20.90.9847

    210.9850

    21.10.9853

    21.20.9856

    21.30.9859

    21.40.9862

    21.50.9864

    21.60.9867

    21.70.9870

    21.80.9872

    21.90.9875

    220.9877

    22.10.9880

    22.20.9882

    22.30.9884

    22.40.9887

    22.50.9889

    22.60.9891

    22.70.9893

    22.80.9895

    22.90.9897

    230.9899

    23.10.9901

    23.20.9903

    23.30.9905

    23.40.9907

    23.50.9909

    23.60.9911

    23.70.9913

    23.80.9914

    23.90.9916

    240.9918

    24.10.9919

    24.20.9921

    24.30.9922

    24.40.9924

    24.50.9926

    24.60.9927

    24.70.9928

    24.80.9930

    24.90.9931

    250.9933

    25.10.9934

    25.20.9935

    25.30.9937

    25.40.9938

    25.50.9939

    25.60.9940

    25.70.9941

    25.80.9943

    25.90.9944

    260.9945

    26.10.9946

    26.20.9947

    26.30.9948

    26.40.9949

    26.50.9950

    26.60.9951

    26.70.9952

    26.80.9953

    26.90.9954

    270.9955

    27.10.9956

    27.20.9957

    27.30.9957

    27.40.9958

    27.50.9959

    27.60.9960

    27.70.9961

    27.80.9962

    27.90.9962

    280.9963

    28.10.9964

    28.20.9964

    28.30.9965

    28.40.9966

    28.50.9967

    28.60.9967

    28.70.9968

    28.80.9968

    28.90.9969

    290.9970

    29.10.9970

    29.20.9971

    29.30.9971

    29.40.9972

    29.50.9973

    29.60.9973

    29.70.9974

    29.80.9974

    29.90.9975

    300.9975

    30.10.9976

    30.20.9976

    30.30.9977

    30.40.9977

    30.50.9978

    30.60.9978

    30.70.9978

    30.80.9979

    30.90.9979

    310.9980

    31.10.9980

    31.20.9981

    31.30.9981

    31.40.9981

    31.50.9982

    31.60.9982

    31.70.9982

    31.80.9983

    31.90.9983

    320.9983

    32.10.9984

    32.20.9984

    32.30.9984

    32.40.9985

    32.50.9985

    32.60.9985

    32.70.9986

    32.80.9986

    32.90.9986

    330.9986

    33.10.9987

    33.20.9987

    33.30.9987

    33.40.9987

    33.50.9988

    33.60.9988

    33.70.9988

    33.80.9988

    33.90.9989

    340.9989

    34.10.9989

    34.20.9989

    34.30.9990

    34.40.9990

    34.50.9990

    34.60.9990

    34.70.9990

    34.80.9991

    34.90.9991

    350.9991

    35.10.9991

    35.20.9991

    35.30.9991

    35.40.9992

    35.50.9992

    35.60.9992

    35.70.9992

    35.80.9992

    35.90.9992

    360.9993

    36.10.9993

    36.20.9993

    36.30.9993

    36.40.9993

    36.50.9993

    36.60.9993

    36.70.9994

    36.80.9994

    36.90.9994

    370.9994

    37.10.9994

    37.20.9994

    37.30.9994

    37.40.9994

    37.50.9994

    37.60.9995

    37.70.9995

    37.80.9995

    37.90.9995

    380.9995

    38.10.9995

    38.20.9995

    38.30.9995

    38.40.9995

    38.50.9995

    38.60.9996

    38.70.9996

    38.80.9996

    38.90.9996

    390.9996

    39.10.9996

    39.20.9996

    39.30.9996

    39.40.9996

    39.50.9996

    39.60.9996

    39.70.9996

    39.80.9997

    39.90.9997

    400.9997

    40.10.9997

    40.20.9997

    40.30.9997

    40.40.9997

    40.50.9997

    40.60.9997

    40.70.9997

    40.80.9997

    40.90.9997

    410.9997

    41.10.9997

    41.20.9997

    41.30.9997

    41.40.9997

    41.50.9998

    41.60.9998

    41.70.9998

    41.80.9998

    41.90.9998

    420.9998

    42.10.9998

    42.20.9998

    42.30.9998

    42.40.9998

    42.50.9998

    42.60.9998

    42.70.9998

    42.80.9998

    42.90.9998

    430.9998

    43.10.9998

    43.20.9998

    43.30.9998

    43.40.9998

    43.50.9998

    43.60.9998

    43.70.9998

    43.80.9998

    43.90.9998

    440.9998

    44.10.9999

    44.20.9999

    44.30.9999

    44.40.9999

    44.50.9999

    44.60.9999

    44.70.9999

    44.80.9999

    44.90.9999

    450.9999

    45.10.9999

    45.20.9999

    45.30.9999

    45.40.9999

    45.50.9999

    45.60.9999

    45.70.9999

    45.80.9999

    45.90.9999

    460.9999

    46.10.9999

    46.20.9999

    46.30.9999

    46.40.9999

    46.50.9999

    46.60.9999

    46.70.9999

    46.80.9999

    46.90.9999

    470.9999

    47.10.9999

    47.20.9999

    47.30.9999

    47.40.9999

    47.50.9999

    47.60.9999

    47.70.9999

    47.80.9999

    47.90.9999

    480.9999

    48.10.9999

    48.20.9999

    48.30.9999

    48.40.9999

    48.50.9999

    48.60.9999

    48.70.9999

    48.80.9999

    48.90.9999

    490.9999

    49.10.9999

    49.20.9999

    49.30.9999

    49.40.9999

    49.50.9999

    49.61.0000

    49.71.0000

    Exp 0,2

    0.0198013207

    0.0392105491

    0.0582354492

    0.0768836311

    0.0951625544

    0.1130795308

    0.1306417275

    0.1478561695

    0.1647297428

    0.181269197

    0.1974811482

    0.2133720814

    0.2289483531

    0.244216194

    0.2591817116

    0.2738508921

    0.2882296035

    0.3023235974

    0.3161385116

    0.3296798722

    0.3429530961

    0.3559634926

    0.368716266

    0.3812165177

    0.3934692479

    0.4054793578

    0.4172516517

    0.4287908387

    0.4401015345

    0.4511882635

    0.4620554608

    0.4727074731

    0.4831485615

    0.4933829026

    0.5034145903

    0.5132476372

    0.5228859769

    0.5323334647

    0.5415938797

    0.5506709263

    0.5595682354

    0.568289366

    0.5768378068

    0.5852169771

    0.5934302287

    0.6014808472

    0.6093720527

    0.617107002

    0.6246887891

    0.6321204467

    0.6394049477

    0.646545206

    0.6535440777

    0.6604043626

    0.6671288047

    0.673720094

    0.6801808671

    0.6865137083

    0.6927211509

    0.6988056779

    0.7047697233

    0.7106156727

    0.7163458646

    0.7219625911

    0.727468099

    0.7328645906

    0.7381542245

    0.7433391167

    0.7484213411

    0.7534029308

    0.7582858785

    0.7630721373

    0.7677636219

    0.7723622089

    0.7768697378

    0.7812880117

    0.785618798

    0.7898638289

    0.7940248026

    0.7981033836

    0.8021012032

    0.8060198608

    0.8098609237

    0.8136259285

    0.8173163813

    0.8209337582

    0.8244795063

    0.8279550439

    0.8313617612

    0.8347010211

    0.8379741592

    0.8411824848

    0.8443272814

    0.8474098068

    0.8504312942

    0.8533929521

    0.8562959653

    0.8591414949

    0.8619306794

    0.8646646343

    0.8673444533

    0.8699712083

    0.8725459501

    0.8750697086

    0.8775434934

    0.8799682939

    0.8823450802

    0.884674803

    0.8869583943

    0.8891967673

    0.8913908177

    0.8935414229

    0.8956494434

    0.8977157222

    0.899741086

    0.9017263449

    0.9036722931

    0.9055797089

    0.9074493554

    0.9092819804

    0.911078317

    0.9128390837

    0.914564985

    0.9162567111

    0.9179149388

    0.9195403315

    0.9211335392

    0.9226951992

    0.9242259364

    0.9257263629

    0.927197079

    0.928638673

    0.9300517215

    0.9314367898

    0.932794432

    0.934125191

    0.9354295992

    0.9367081784

    0.9379614401

    0.9391898855

    0.9403940061

    0.9415742835

    0.9427311898

    0.9438651879

    0.9449767313

    0.9460662647

    0.9471342239

    0.9481810361

    0.94920712

    0.9502128861

    0.9511987367

    0.9521650662

    0.9531122611

    0.9540407002

    0.954950755

    0.9558427896

    0.9567171607

    0.9575742181

    0.9584143046

    0.9592377563

    0.9600449025

    0.9608360662

    0.9616115638

    0.9623717056

    0.9631167955

    0.9638471317

    0.9645630062

    0.9652647055

    0.9659525102

    0.9666266955

    0.967287531

    0.9679352811

    0.9685702048

    0.9691925563

    0.9698025844

    0.9704005331

    0.9709866416

    0.9715611444

    0.9721242713

    0.9726762476

    0.973217294

    0.9737476269

    0.9742674586

    0.9747769969

    0.9752764456

    0.9757660047

    0.9762458698

    0.9767162329

    0.9771772823

    0.9776292022

    0.9780721736

    0.9785063735

    0.9789319757

    0.9793491504

    0.9797580645

    0.9801588816

    0.9805517619

    0.9809368627

    0.981314338

    0.9816843388

    0.9820470131

    0.9824025059

    0.9827509595

    0.9830925133

    0.9834273039

    0.9837554652

    0.9840771284

    0.9843924223

    0.9847014729

    0.985004404

    0.9853013366

    0.9855923895

    0.9858776793

    0.9861573199

    0.9864314232

    0.9867000989

    0.9869634546

    0.9872215954

    0.9874746247

    0.9877226436

    0.9879657515

    0.9882040455

    0.988437621

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    Plan2

    0

    0.0951604666

    0.1812654188

    0.2591765835

    0.3296736855

    0.3934622504

    0.4511806656

    0.5034065696

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    0.9979702752

    0.9981634247

    0.9983381941

    0.9984963323

    0.998639422

    0.9987688953

    0.9988860478

    0.998992052

    0.9990879688

    N

    xf(x)xf(x)

    -1-0.2210.10.0198

    -0.5-0.1050.20.0392

    000.30.0582

    0.50.09520.40.0769

    10.18130.50.0952

    1.50.25920.60.1131

    20.32970.70.1306

    2.50.39350.80.1479

    30.45120.90.1647

    3.50.503410.1813

    40.55071.10.1975

    4.50.59341.20.2134

    50.63211.30.2289

    5.50.66711.40.2442

    60.69881.50.2592

    6.50.72751.60.2739

    70.75341.70.2882

    7.50.77691.80.3023

    80.79811.90.3161

    8.50.817320.3297

    90.83472.10.3430

    9.50.85042.20.3560

    100.86472.30.3687

    10.50.87752.40.3812

    110.88922.50.3935

    11.50.89972.60.4055

    120.90932.70.4173

    12.50.91792.80.4288

    130.92572.90.4401

    13.50.932830.45120.3469

    140.93923.10.4621

    14.50.94503.20.4727

    150.95023.30.4831

    15.50.95493.40.4934

    160.95923.50.5034

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    170.96663.70.5229

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    180.97273.90.5416

    18.50.975340.5507

    190.97764.10.5596

    19.50.97984.20.5683

    200.98174.30.5768

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    210.98504.50.5934

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    220.98774.70.6094

    22.50.98894.80.6171

    230.98994.90.6247

    23.50.990950.6321

    240.99185.10.6394

    24.50.99265.20.6465

    250.99335.30.6535

    25.50.99395.40.6604

    260.99455.50.6671

    26.50.99505.60.6737

    270.99555.70.6802

    27.50.99595.80.6865

    280.99635.90.6927

    28.50.996760.6988

    290.99706.10.7048

    29.50.99736.20.7106

    300.99756.30.7163

    30.50.99786.40.7220

    310.99806.50.7275

    31.50.99826.60.7329

    320.99836.70.7382

    32.50.99856.80.7433

    330.99866.90.7484

    33.50.998870.7534

    340.99897.10.7583

    34.50.99907.20.7631

    350.99917.30.7678

    35.50.99927.40.7724

    360.99937.50.7769

    36.50.99937.60.7813

    370.99947.70.7856

    37.50.99947.80.7899

    380.99957.90.7940

    38.50.999580.7981

    390.99968.10.8021

    39.50.99968.20.8060

    400.99978.30.8099

    40.50.99978.40.8136

    410.99978.50.8173

    41.50.99988.60.8209

    420.99988.70.8245

    42.50.99988.80.8280

    430.99988.90.8314

    43.50.999890.8347

    440.99989.10.8380

    44.50.99999.20.8412

    450.99999.30.8443

    45.50.99999.40.8474

    460.99999.50.8504

    46.50.99999.60.8534

    470.99999.70.8563

    47.50.99999.80.8591

    480.99999.90.8619

    48.50.9999100.8647

    490.999910.10.8673

    49.50.999910.20.8700

    501.000010.30.8725

    10.40.8751

    10.50.8775

    10.60.8800

    10.70.8823

    10.80.8847

    10.90.8870

    110.8892

    11.10.8914

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    11.70.9037

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    11.90.9074

    120.9093

    12.10.9111

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    130.9257

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    150.9502

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    180.9727

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    240.9918

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    250.9933

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    260.9945

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    270.9955

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    280.9963

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    290.9970

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    300.9975

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    310.9980

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    320.9983

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    43.90.9998

    440.9998

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    44.60.9999

    44.70.9999

    44.80.9999

    44.90.9999

    450.9999

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    45.20.9999

    45.30.9999

    45.40.9999

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    45.60.9999

    45.70.9999

    45.80.9999

    45.90.9999

    460.9999

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    46.40.9999

    46.50.9999

    46.60.9999

    46.70.9999

    46.80.9999

    46.90.9999

    470.9999

    47.10.9999

    47.20.9999

    47.30.9999

    47.40.9999

    47.50.9999

    47.60.9999

    47.70.9999

    47.80.9999

    47.90.9999

    480.9999

    48.10.9999

    48.20.9999

    48.30.9999

    48.40.9999

    48.50.9999

    48.60.9999

    48.70.9999

    48.80.9999

    48.90.9999

    490.9999

    49.10.9999

    49.20.9999

    49.30.9999

    49.40.9999

    49.50.9999

    49.61.0000

    49.71.0000

    N

    0.0198013207

    0.0392105491

    0.0582354492

    0.0768836311

    0.0951625544

    0.1130795308

    0.1306417275

    0.1478561695

    0.1647297428

    0.181269197

    0.1974811482

    0.2133720814

    0.2289483531

    0.244216194

    0.2591817116

    0.2738508921

    0.2882296035

    0.3023235974

    0.3161385116

    0.3296798722

    0.3429530961

    0.3559634926

    0.368716266

    0.3812165177

    0.3934692479

    0.4054793578

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