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Mtodos Estatsticos II
Prof.: Manuel Rosa de Oliveira Lino
Variveis Aleatrias
Resumo Unidade 3
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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
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Uma varivel aleatria pode ser explicada por uma funo que associa a cada elemento sS um nmero real X(s).Conceito
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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:
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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:
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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:
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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:
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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
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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
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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:
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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)
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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)
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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)
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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
2SNSSSSSSNSSSSSSSSSSS
3SSSSSSSSSSSSSSSSSNSS
4SSNSSNSSSSNSSNSSNSSS
5NSSSNSSSNSSSSNSSSSNS
6SNSSSNSNSNSNSSNSSNSS
7SSSNSNSSNSNSSSSNSSNS
8SNSSSNSSSNSSNSSSNSSN
9NSNSNSNSNSSNSSNSSNSN
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
10.38150.47310.011.000
20.89150.88461.010.500
30.08050.81021.010.333
40.57550.98121.010.250
50.62820.34390.020.400
60.66360.82631.020.333
70.55090.71611.020.286
80.23210.75001.020.250
90.83800.20420.030.333
100.69550.48970.040.400
110.77010.90241.040.364
120.91750.20060.050.417
130.54100.72461.050.385
140.58980.49880.060.429
150.99450.69411.060.400
160.54330.80551.060.375
170.95990.05140.070.412
180.15000.89171.070.389
190.39660.07170.080.421
200.08470.82251.080.400
210.88640.40710.090.429
220.85010.88061.090.409
230.44590.87991.090.391
240.41290.47650.0100.417
250.66160.42330.0110.440
260.43890.37100.0120.462
270.64380.01140.0130.481
280.07290.01890.0140.500
290.72990.50901.0140.483
300.62340.49130.0150.500
310.53150.59691.0150.484
320.29530.32770.0160.500
330.73640.74621.0160.485
340.78760.54931.0160.471
350.29170.18600.0170.486
360.17240.51831.0170.472
370.59230.37450.0180.486
380.46030.75301.0180.474
390.38780.27310.0190.487
400.85490.72551.0190.475
410.80740.72711.0190.463
420.50300.92751.0190.452
430.14180.56621.0190.442
440.19930.65951.0190.432
450.17030.31430.0200.444
460.87230.30760.0210.457
470.46370.93751.0210.447
480.99050.85901.0210.438
490.60170.82021.0210.429
500.43650.15080.0220.440
510.06570.93961.0220.431
520.17790.22850.0230.442
530.58820.67481.0230.434
540.45130.03970.0240.444
550.52310.69871.0240.436
560.75070.30530.0250.446
570.14960.87961.0250.439
580.68010.21480.0260.448
590.83830.40620.0270.458
600.12670.88471.0270.450
610.46070.01570.0280.459
620.13190.75891.0280.452
630.47480.66261.0280.444
640.74040.58871.0280.438
650.44530.56501.0280.431
660.14220.80121.0280.424
670.11140.42490.0290.433
680.13840.33240.0300.441
690.09470.98981.0300.435
700.84980.84081.0300.429
710.01800.55241.0300.423
720.61970.38470.0310.431
730.40500.57261.0310.425
740.25220.32540.0320.432
750.45500.02060.0330.440
760.10260.39620.0340.447
770.60360.93481.0340.442
780.70210.71731.0340.436
790.65520.81521.0340.430
800.83030.91841.0340.425
810.43510.98631.0340.420
820.41190.53711.0340.415
830.06540.33160.0350.422
840.03260.48230.0360.429
850.23660.49140.0370.435
860.66930.59761.0370.430
870.74180.96661.0370.425
880.07220.35180.0380.432
890.43640.29890.0390.438
900.89280.82631.0390.433
910.88730.64671.0390.429
920.56970.30830.0400.435
930.30330.53051.0400.430
940.60490.40830.0410.436
950.36660.63981.0410.432
960.72980.82841.0410.427
970.31910.87791.0410.423
980.75310.63031.0410.418
990.64940.05740.0420.424
1000.00770.13240.0430.430
1010.22800.85321.0430.426
1020.34970.82541.0430.422
1030.10150.22920.0440.427
1040.07610.00330.0450.433
1050.55750.57611.0450.429
1060.72580.58771.0450.425
1070.65640.25550.0460.430
1080.45890.27180.0470.435
1090.70380.33530.0480.440
1100.29250.73661.0480.436
1110.68810.26830.0490.441
1120.46210.14630.0500.446
1130.43760.14410.0510.451
1140.46880.91691.0510.447
1150.87830.71101.0510.443
1160.31310.89031.0510.440
1170.25970.42550.0520.444
1180.44650.04040.0530.449
1190.82870.45670.0540.454
1200.89880.90551.0540.450
1210.75960.53021.0540.446
1220.67270.71211.0540.443
1230.88000.31250.0550.447
1240.47630.20000.0560.452
1250.83740.61001.0560.448
1260.68000.31780.0570.452
1270.77320.32320.0580.457
1280.45080.76951.0580.453
1290.47360.58041.0580.450
1300.10100.87791.0580.446
1310.86430.67051.0580.443
1320.01050.87141.0580.439
1330.71660.53461.0580.436
1340.08640.96541.0580.433
1350.10590.54871.0580.430
1360.54570.64831.0580.426
1370.22030.20310.0590.431
1380.19490.39470.0600.435
1390.61840.59801.0600.432
1400.82440.26640.0610.436
1410.96600.49900.0620.440
1420.54410.06680.0630.444
1430.35040.32550.0640.448
1440.52690.45030.0650.451
1450.96720.25920.0660.455
1460.54220.61511.0660.452
1470.50070.77591.0660.449
1480.77370.37840.0670.453
1490.14930.65451.0670.450
1500.12080.81481.0670.447
1510.68380.71151.0670.444
1520.65710.75511.0670.441
1530.95500.75201.0670.438
1540.58900.29130.0680.442
1550.69930.30580.0690.445
1560.07400.68901.0690.442
1570.20990.01630.0700.446
1580.57800.53661.0700.443
1590.62020.26210.0710.447
1600.71290.86801.0710.444
1610.91660.56631.0710.441
1620.74530.47690.0720.444
1630.77950.79571.0720.442
1640.48140.67051.0720.439
1650.67170.85661.0720.436
1660.83560.55281.0720.434
1670.24200.10250.0730.437
1680.88410.18490.0740.440
1690.37430.66591.0740.438
1700.18980.63511.0740.435
1710.25950.45890.0750.439
1720.63840.69601.0750.436
1730.68910.39130.0760.439
1740.85400.13790.0770.443
1750.82770.17040.0780.446
1760.26990.46370.0790.449
1770.99440.76191.0790.446
1780.07600.88981.0790.444
1790.41690.49710.0800.447
1800.84180.54061.0800.444
1810.41330.33680.0810.448
1820.84030.50271.0810.445
1830.12190.59041.0810.443
1840.83290.17920.0820.446
1850.95880.79601.0820.443
1860.43190.27060.0830.446
1870.34600.67221.0830.444
1880.15510.75401.0830.441
1890.26540.10840.0840.444
1900.24690.35180.0850.447
1910.11250.44070.0860.450
1920.96210.37130.0870.453
1930.36040.72901.0870.451
1940.64260.96461.0870.448
1950.63460.52271.0870.446
1960.12440.06040.0880.449
1970.89450.58061.0880.447
1980.93640.33330.0890.449
1990.65810.36050.0900.452
2000.14330.85751.0900.450
2010.34970.53221.0900.448
2020.21890.36730.0910.450
2030.55000.42100.0920.453
2040.77250.24530.0930.456
2050.20790.87291.0930.454
2060.53210.99211.0930.451
2070.02580.11530.0940.454
2080.01600.89011.0940.452
2090.15340.26190.0950.455
2100.69090.71201.0950.452
2110.20600.16740.0960.455
2120.59590.31900.0970.458
2130.38770.07800.0980.460
2140.49790.34170.0990.463
2150.25320.00910.01000.465
2160.67330.18470.01010.468
2170.69350.80421.01010.465
2180.31320.15300.01020.468
2190.87070.73511.01020.466
2200.48420.11710.01030.468
2210.29220.12750.01040.471
2220.17350.90881.01040.468
2230.46030.67691.01040.466
2240.78470.52601.01040.464
2250.37830.59231.01040.462
2260.20180.08140.01050.465
2270.82680.12650.01060.467
2280.15520.60231.01060.465
2290.87110.11430.01070.467
2300.49060.27440.01080.470
2310.45480.57741.01080.468
2320.28790.16960.01090.470
2330.44010.49200.01100.472
2340.98060.80161.01100.470
2350.06090.36960.01110.472
2360.11640.73421.01110.470
2370.13650.29740.01120.473
2380.95350.00530.01130.475
2390.17870.14840.01140.477
2400.75460.97431.01140.475
2410.52550.14480.01150.477
2420.26500.59821.01150.475
2430.98190.48200.01160.477
2440.61000.75961.01160.475
2450.76450.25780.01170.478
2460.70710.28270.01180.480
2470.27460.07770.01190.482
2480.91330.32840.01200.484
2490.73310.78191.01200.482
2500.32590.03700.01210.484
2510.82850.02540.01220.486
2520.83000.49820.01230.488
2530.61440.86741.01230.486
2540.20510.22710.01240.488
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2560.18210.59531.01240.484
2570.79670.97951.01240.482
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2590.60860.07180.01250.483
2600.04950.45960.01260.485
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2650.46390.08550.01280.483
2660.80300.91151.01280.481
2670.14500.57901.01280.479
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2690.35840.57281.01280.476
2700.65580.56441.01280.474
2710.02350.18060.01290.476
2720.78290.13360.01300.478
2730.08200.61131.01300.476
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2780.66380.03830.01310.471
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2800.69510.24560.01330.475
2810.92110.58801.01330.473
2820.53090.10130.01340.475
2830.44160.43220.01350.477
2840.77370.19720.01360.479
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2890.82100.91981.01380.478
2900.59680.32790.01390.479
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2920.45320.47690.01400.479
2930.49030.78201.01400.478
2940.75920.13280.01410.480
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2990.47790.20750.01440.482
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3010.43830.42430.01460.485
3020.47520.08150.01470.487
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3050.54450.08310.01500.492
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3110.31980.20400.01530.492
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3130.74270.26550.01540.492
3140.03490.53861.01540.490
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3160.71010.15520.01550.491
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3190.94150.24560.01570.492
3200.48430.71341.01570.491
3210.12620.88571.01570.489
3220.35950.40180.01580.491
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3690.13360.42520.01830.496
3700.67650.39570.01840.497
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3730.01580.22220.01850.496
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3810.74630.28980.01910.501
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3840.06570.28620.01920.500
3850.90400.39770.01930.501
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3870.78250.61121.01930.499
3880.32520.47450.01940.500
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3910.92180.51141.01940.496
3920.79250.23570.01950.497
3930.15500.09520.01960.499
3940.26470.57961.01960.497
3950.67830.92501.01960.496
3960.63060.80381.01960.495
3970.94480.29770.01970.496
3980.64060.55371.01970.495
3990.39430.31720.01980.496
4000.48140.10710.01990.498
4010.69710.84461.01990.496
4020.89120.60931.01990.495
4030.09490.35360.02000.496
4040.72060.90371.02000.495
4050.38960.88861.02000.494
4060.93540.21290.02010.495
4070.16590.10600.02020.496
4080.73740.55781.02020.495
4090.05810.56351.02020.494
4100.30610.16550.02030.495
4110.03060.94151.02030.494
4120.36100.03140.02040.495
4130.78310.58991.02040.494
4140.89520.41570.02050.495
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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
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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
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120.90932.70.4173
12.50.91792.80.4288
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13.50.932830.45120.3469
140.93923.10.4621
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16.50.96313.60.5132
170.96663.70.5229
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180.97273.90.5416
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190.97764.10.5596
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220.98774.70.6094
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23.50.990950.6321
240.99185.10.6394
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25.50.99395.40.6604
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38.50.999580.7981
390.99968.10.8021
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42.50.99988.80.8280
430.99988.90.8314
43.50.999890.8347
440.99989.10.8380
44.50.99999.20.8412
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480.99999.90.8619
48.50.9999100.8647
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501.000010.30.8725
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49.61.0000
49.71.0000
N
0.0198013207
0.0392105491
0.0582354492
0.0768836311
0.0951625544
0.1130795308
0.1306417275
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