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Bioestadística FMCS Reus Curs 2013-14 URV 1
Exercicis
Variable aleatòria
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Bioestadística FMCS Reus Curs 2013-14 URV 2
Propietats de l'esperança i la variància d'una variable aleatòria discreta
Propietats de l'esperança
)X(kE)kX(E
)Y(E)X(E)YX(E
k)k(E
Propietats de la Variància
)X(Vk=)kX(V
0=)k(V2
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Bioestadística FMCS Reus Curs 2013-14 URV 3
Exercici 1
a) E(Y) = E(3X+6) = E(3X)+E(6) = 3E(X) + 6 = 3·3 + 6 = 15
X: E(X)=3 V(X)=2
Y=3X+6
b) V(Y) = V(3X+6) = V(3X)+V(6) = 9V(X) = 9·2 = 18
c) DE(Y) = = = 4’24 V(Y) 18
d) CV(Y) = DE(Y)/E(Y) = 4’24/15 = 0,28
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Bioestadística FMCS Reus Curs 2013-14 URV 4
Variables aleatòries discretes
Funció de probabilitat: Es una funció que associa a cada resultat
possible de la variable discreta X la probabilitat de aquest resultat
P(x)=P(X=x)
Funció de distribució de probabilitat F(x)=P(X≤x)
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Bioestadística FMCS Reus Curs 2013-14 URV 5
Exercici 2
x 0 1 2 3 4 5 6
F(x) 0’1 0’2 0’3 0’6 0’8 0’9 1
b) P(X≤3) = F(3) = 0’6
c) P(X≥1) = 1 - P(X=0) = 1- 0’1 =0’9
d) P(2 ≤ X ≤ 4) = P(X ≤ 4) - P(X ≤ 1) = F(4) – F(1) = 0’8 – 0’2 = 0’6
x 0 1 2 3 4 5 6
F(x) 0’1 0’2 0’3 0’6 0’8 0’9 1
P(x) 0’1 0’1 0’1 0’3 0’2 0’1 0’1
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Bioestadística FMCS Reus Curs 2013-14 URV 6
Exercici 2
)x(Px=)X(Extot
∑
x 0 1 2 3 4 5 6
F(x) 0’1 0’2 0’3 0’6 0’8 0’9 1
P(x) 0’1 0’1 0’1 0’3 0’2 0’1 0’1
E(X) = 0·0’1 + 1·0’1 + 2·0’1 + 3·0’3 + 4·0’2 + 5·0’1 + 6·0’1 = 3’1
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Bioestadística FMCS Reus Curs 2013-14 URV 7
Qualsevol variable aleatòria X que representi el nombre d'èxits en n proves idèntiques i independents, amb probabilitat d'èxit p constant d'una prova a l'altre, se'n diu una variable aleatòria binomial amb paràmetres n i p.
X~B(n,p)
Variable aleatòria binomial
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Bioestadística FMCS Reus Curs 2013-14 URV 8
Variable aleatoria binomial
Funció de probabilitat
xnxnx )p(p)xX(P --1
pn)X(E)p,n(B:X ·
Esperança o mitjana
ppn)X(V -1 ·
Variància
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Bioestadística FMCS Reus Curs 2013-14 URV 99
Exercici 3
Efectiu 90% de les vegades (p=0.90) S’experimenta en 4 individus (n=4)
X: Nombre de pacients amb millora X~B(4,0’9)
Funció de probabilitat xnxn
x )p(p)xX(P --1
TAULES
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Bioestadística FMCS Reus Curs 2013-14 URV 10Bioestadística FMCS Reus Curs 2012-13 URV 10
Probabilitats binomials exactes
P{X = x} =
x
npx (1 – p)n – x n = 1, 2, ...,10
n x \ p .01 .05 .10 .15 .20 .25 .30 1/3 .35 .40 .45 .49 .50 2 0 .9801 .9025 .8100 .7225 .6400 .5625 .4900 .4444 .4225 .3600 .3025 .2601 .2500
1 .0198 .9050 .1800 .2550 .3200 .3750 .4200 .4444 .4550 .4800 .4950 .4998 .5000 2 .0001 .0025 .0100 .0225 .0400 .0625 .0900 .1111 .1225 .1600 .2050 .2401 .2500 3 0 .9703 .8574 .7290 .6141 .5120 .4219 .3430 .2963 .2746 .2160 .1644 .1327 .1250
1 .0294 .1354 .2430 .3251 .3840 .4219 .4410 .4444 .4436 .4320 .4084 .3823 .3750 2 .0003 .0071 .0270 .0574 .0960 .1406 .1890 .2222 .2389 .2880 .3341 .3674 .3750 3 .0000 .0001 .0010 .0034 .0080 .0156 .0270 .0370 .0429 .0640 .0911 .1176 .1250 4 0 .9606 .8145 .6561 .5220 .4096 .3164 .2401 .1975 .1785 .1296 .0915 .0677 .0625
1 .0388 .1715 .2916 .3685 .4096 .4219 .4116 .3951 .3845 .3456 .2995 .2600 .2500 2 .0006 .0135 .0486 .0975 .1536 .2109 .2646 .2963 .3105 .3456 .3675 .3747 .3750 3 .0000 .0005 .0036 .0115 .0256 .0469 .0756 .0988 .1115 .1536 .2005 .2400 .2500 4 .0000 .0000 .0001 .0005 .0016 .0039 .0081 .0123 .0150 .0256 .0410 .0576 .0625 5 0 .9510 .7738 .5905 .4437 .3277 .2373 .1681 .1317 .1160 .0778 .0503 .0345 .0312
1 .0480 .2036 .3280 .3915 .4096 .3955 .3602 .3292 .3124 .2592 .2059 .1657 .1562 2 .0010 .0214 .0729 .1582 .2043 .2637 .3087 .3292 .3364 .3456 .3369 .3185 .3125 3 .0000 .0011 .0081 .0244 .0512 .0879 .1323 .1646 .1811 .2304 .2757 .3060 .3125 4 .0000 .0000 .0004 .0022 .0064 .0145 .0284 .0412 .0488 .0768 .1128 .1470 .1562 5 .0000 .0000 .0000 .0001 .0003 .0010 .0024 .0041 .0053 .0102 .0185 .0283 .0312 6 0 .9415 .7351 .5314 .3771 .2621 .1780 .1176 .0872 .0754 .0467 .0277 .0176 .0156
1 .0571 .2321 .3543 .3993 .3932 .3560 .3025 .2634 .2437 .1866 .1359 .1014 .0938 2 .0014 .0305 .0984 .1762 .2458 .2966 .3241 .3292 .3280 .3110 .2780 .2437 .2344 3 .0000 .0021 .0146 .0415 .0819 .1312 .1852 .2195 .2355 .2765 .3032 .3121 .3125 4 .0000 .0001 .0012 .0055 .0154 .0330 .0592 .0823 .0951 .1382 .1861 .2249 .2344 5 .0000 .0000 .0001 .0004 .0015 .0044 .0102 .0165 .0205 .0369 .0609 .0864 .0938 6 .0000 .0000 .0000 .0000 .0002 .0002 .0007 .0014 .0018 .0041 .0083 .0139 .0156 7 0 .9321 .6983 .4783 .3206 .2097 .1335 .0824 .0585 .0490 .0280 .0152 .0090 .0078
1 .0659 .2575 .3720 .3960 .3670 .3115 .2471 .2048 .1848 .1306 .0872 .0603 .0547 2 .0020 .0406 .1240 .2097 .2753 .3115 .3177 .3073 .2985 .2613 .2140 .1740 .1641 3 .0000 .0036 .0230 .0617 .1147 .1730 .2269 .2561 .2679 .2903 .1918 .2786 .2734 4 .0000 .0002 .0026 .0109 :0287 .0577 .0972 .1280 .1442 .1935 .2388 .2676 .2734 5 .0000 .0000 .0002 .0012 .0043 .0115 .0250 .0384 .0466 .0774 .1172 .1543 .1641 6 .0000 .0000 .0000 .0001 .0004 .0013 .0036 .0064 .0084 .0172 .0320 .0494 .0547 7 .0000 .0000 .0000 .0000 .0000 .0001 .0002 .0005 .0006 .0016 .0037 .0068 .0078 8 0 .9227 .6634 .4305 .2725 .1678 .1001 .0576 .0390 .0319 .0168 .0084 .0046 .0039
1 .0746 .2793 .3826 .3847 .3355 .2670 .1977 .1373 .1373 .0896 .0548 .0352 .0312 2 .0001 .0515 .1488 .2376 .2936 .3115 .2965 .2587 .2587 .2090 .1569 .1183 .1 094 3 .0000 .0054 .0351 .0839 .1468 .2076 .2541 .2786 .2786 .2787 .2568 .2273 .2188 4 .0000 .0004 .0046 .0185 .0459 .0865 .1361 .1875 .1875 .2322 .2627 .2730 .2734 5 .0000 .0000 .0004 .0026 .0092 .0251 .0467 .0808 .0808 .1239 .1719 .2098 .2188 6 .0000 .0000 .0000 .0002 .0011 .0038 .0100 .0217 .0217 .0413 .0703 .1008 .1094 7 .0000 .0000 .0000 .0000 .0001 .0004 .0012 .0033 .0033 .0079 .0164 .0277 .0312
8 .0000 .0000 .0000 .0000 .0000 .0000 .0001 .0002 .0002 .0007 .0017 .0033 .0039
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Bioestadística FMCS Reus Curs 2013-14 URV 11
Exercici 3
Efectiu 90% de les vegades (p=0.90) S’experimenta en 4 individus (n=4)
X: Nombre de pacients amb millora X~B(4,0’9)
Funció de probabilitat xnxn
x )p(p)xX(P --1
NO EXISTEIX TAULA: X~B(4,0’9)
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Bioestadística FMCS Reus Curs 2013-14 URV 12
Exercici 3
Efectiu 90% de les vegades (p=0.90) S’experimenta en 4 individus (n=4)
X: Nombre de pacients amb millora X~B(4,0’9)
Y: Nombre de pacients que no milloren Y~B(4,0’1)
X=4-Y
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Bioestadística FMCS Reus Curs 2013-14 URV 13Bioestadística FMCS Reus Curs 2012-13 URV 13
Probabilitats binomials exactes
P{X = x} =
x
npx (1 – p)n – x n = 1, 2, ...,10
n x \ p .01 .05 .10 .15 .20 .25 .30 1/3 .35 .40 .45 .49 .50 2 0 .9801 .9025 .8100 .7225 .6400 .5625 .4900 .4444 .4225 .3600 .3025 .2601 .2500
1 .0198 .9050 .1800 .2550 .3200 .3750 .4200 .4444 .4550 .4800 .4950 .4998 .5000 2 .0001 .0025 .0100 .0225 .0400 .0625 .0900 .1111 .1225 .1600 .2050 .2401 .2500 3 0 .9703 .8574 .7290 .6141 .5120 .4219 .3430 .2963 .2746 .2160 .1644 .1327 .1250
1 .0294 .1354 .2430 .3251 .3840 .4219 .4410 .4444 .4436 .4320 .4084 .3823 .3750 2 .0003 .0071 .0270 .0574 .0960 .1406 .1890 .2222 .2389 .2880 .3341 .3674 .3750 3 .0000 .0001 .0010 .0034 .0080 .0156 .0270 .0370 .0429 .0640 .0911 .1176 .1250 4 0 .9606 .8145 .6561 .5220 .4096 .3164 .2401 .1975 .1785 .1296 .0915 .0677 .0625
1 .0388 .1715 .2916 .3685 .4096 .4219 .4116 .3951 .3845 .3456 .2995 .2600 .2500 2 .0006 .0135 .0486 .0975 .1536 .2109 .2646 .2963 .3105 .3456 .3675 .3747 .3750 3 .0000 .0005 .0036 .0115 .0256 .0469 .0756 .0988 .1115 .1536 .2005 .2400 .2500 4 .0000 .0000 .0001 .0005 .0016 .0039 .0081 .0123 .0150 .0256 .0410 .0576 .0625 5 0 .9510 .7738 .5905 .4437 .3277 .2373 .1681 .1317 .1160 .0778 .0503 .0345 .0312
1 .0480 .2036 .3280 .3915 .4096 .3955 .3602 .3292 .3124 .2592 .2059 .1657 .1562 2 .0010 .0214 .0729 .1582 .2043 .2637 .3087 .3292 .3364 .3456 .3369 .3185 .3125 3 .0000 .0011 .0081 .0244 .0512 .0879 .1323 .1646 .1811 .2304 .2757 .3060 .3125 4 .0000 .0000 .0004 .0022 .0064 .0145 .0284 .0412 .0488 .0768 .1128 .1470 .1562 5 .0000 .0000 .0000 .0001 .0003 .0010 .0024 .0041 .0053 .0102 .0185 .0283 .0312 6 0 .9415 .7351 .5314 .3771 .2621 .1780 .1176 .0872 .0754 .0467 .0277 .0176 .0156
1 .0571 .2321 .3543 .3993 .3932 .3560 .3025 .2634 .2437 .1866 .1359 .1014 .0938 2 .0014 .0305 .0984 .1762 .2458 .2966 .3241 .3292 .3280 .3110 .2780 .2437 .2344 3 .0000 .0021 .0146 .0415 .0819 .1312 .1852 .2195 .2355 .2765 .3032 .3121 .3125 4 .0000 .0001 .0012 .0055 .0154 .0330 .0592 .0823 .0951 .1382 .1861 .2249 .2344 5 .0000 .0000 .0001 .0004 .0015 .0044 .0102 .0165 .0205 .0369 .0609 .0864 .0938 6 .0000 .0000 .0000 .0000 .0002 .0002 .0007 .0014 .0018 .0041 .0083 .0139 .0156 7 0 .9321 .6983 .4783 .3206 .2097 .1335 .0824 .0585 .0490 .0280 .0152 .0090 .0078
1 .0659 .2575 .3720 .3960 .3670 .3115 .2471 .2048 .1848 .1306 .0872 .0603 .0547 2 .0020 .0406 .1240 .2097 .2753 .3115 .3177 .3073 .2985 .2613 .2140 .1740 .1641 3 .0000 .0036 .0230 .0617 .1147 .1730 .2269 .2561 .2679 .2903 .1918 .2786 .2734 4 .0000 .0002 .0026 .0109 :0287 .0577 .0972 .1280 .1442 .1935 .2388 .2676 .2734 5 .0000 .0000 .0002 .0012 .0043 .0115 .0250 .0384 .0466 .0774 .1172 .1543 .1641 6 .0000 .0000 .0000 .0001 .0004 .0013 .0036 .0064 .0084 .0172 .0320 .0494 .0547 7 .0000 .0000 .0000 .0000 .0000 .0001 .0002 .0005 .0006 .0016 .0037 .0068 .0078 8 0 .9227 .6634 .4305 .2725 .1678 .1001 .0576 .0390 .0319 .0168 .0084 .0046 .0039
1 .0746 .2793 .3826 .3847 .3355 .2670 .1977 .1373 .1373 .0896 .0548 .0352 .0312 2 .0001 .0515 .1488 .2376 .2936 .3115 .2965 .2587 .2587 .2090 .1569 .1183 .1 094 3 .0000 .0054 .0351 .0839 .1468 .2076 .2541 .2786 .2786 .2787 .2568 .2273 .2188 4 .0000 .0004 .0046 .0185 .0459 .0865 .1361 .1875 .1875 .2322 .2627 .2730 .2734 5 .0000 .0000 .0004 .0026 .0092 .0251 .0467 .0808 .0808 .1239 .1719 .2098 .2188 6 .0000 .0000 .0000 .0002 .0011 .0038 .0100 .0217 .0217 .0413 .0703 .1008 .1094 7 .0000 .0000 .0000 .0000 .0001 .0004 .0012 .0033 .0033 .0079 .0164 .0277 .0312
8 .0000 .0000 .0000 .0000 .0000 .0000 .0001 .0002 .0002 .0007 .0017 .0033 .0039
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Bioestadística FMCS Reus Curs 2013-14 URV 1414
Exercici 3
Y X P(x) F(x)
4 0 0,0001 0,0001
3 1 0,0036 0,0037
2 2 0,0486 0,0523
1 3 0,2916 0,3439
0 4 0,6561 1
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Bioestadística FMCS Reus Curs 2013-14 URV 15
Exercici 3
pn)X(E)p,n(B:X ·
Esperança o mitjana
ppn)X(V -1 · Variància
X: B(4,0,9)
E(X) = np = 4*0,9 = 3,6
V(X) = np(1-p) = 4*0,9*0,1 = 0,36
DE(X) = √V(X) = √ 0.36 = 0,6
CV(X) = DE(X)/E(X) = 0,6/3,6 = 0.1667
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Bioestadística FMCS Reus Curs 2013-14 URV 16
Exercici 4
Meningitis: Mortalitat 30% Supervivència 70% 9 nens amb meningitis (n=9)
X: Nombre de nens que sobreviuen de meningitis X~B(9,0’7)
NO EXISTEIX TAULA: X~B(9,0’7)
Y: Nombre de nens que moren de meningitis Y~B(9,0’3)
X=9-Y
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Bioestadística FMCS Reus Curs 2013-14 URV 1717
Exercici 4
X: Nombre de nens que sobreviuen de meningitisY: Nombre de nens que moren de meningitisY~B(9,0’3)
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Bioestadística FMCS Reus Curs 2013-14 URV 1818
Exercici 4
Y~B(9,0’3) Y P(Y=y)
0 0,0404
1 0,1556
2 0,2668
3 0,2668
4 0,1715
5 0,0735
6 0,0210
7 0,0039
8 0,0004
9 0,0000
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Bioestadística FMCS Reus Curs 2013-14 URV 1919
Exercici 4
a) P(X≥5) = P(Y≤4) = P(Y=0)+P(Y=1)+P(Y=2)+P(Y=3)+P(Y=4) =
0,0404 + 0,1556 + 0,2668 + 0,2668 + 0,1715 = 0,9011
X: Nombre de nens que sobreviuen de meningitisY: Nombre de nens que moren de meningitisY~B(9,0’3)
b) P(X=9) = P(Y=0) = 0,0404
c) P(6≤X≤8) = P(1≤Y≤3) = P(Y=1)+P(Y=2)+P(Y=3) =
0,1556 + 0,2668 + 0,2668 = 0,6892
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Bioestadística FMCS Reus Curs 2013-14 URV 20
Exercici 5
N=8 P=0’9 X: Nombre de persones sense crisi de migranya X~B(8,0’9)
N=8 P=0’1 Y: Nombre de persones amb crisi de migranya Y~B(8,0’1)
O
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Bioestadística FMCS Reus Curs 2013-14 URV 21
Exercici 5
X: Nombre de persones sense crisi de migranyaY: Nombre de persones amb crisi de migranyaY~B(8,0’1)
a) P(X≥7) = P(Y≤1) = P(Y=1)+P(Y=0)
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Bioestadística FMCS Reus Curs 2013-14 URV 22
Probabilitats binomials exactes
P{X = x} =
x
npx (1 – p)n – x n = 1, 2, ...,10
n x \ p .01 .05 .10 .15 .20 .25 .30 1/3 .35 .40 .45 .49 .50 2 0 .9801 .9025 .8100 .7225 .6400 .5625 .4900 .4444 .4225 .3600 .3025 .2601 .2500
1 .0198 .9050 .1800 .2550 .3200 .3750 .4200 .4444 .4550 .4800 .4950 .4998 .5000 2 .0001 .0025 .0100 .0225 .0400 .0625 .0900 .1111 .1225 .1600 .2050 .2401 .2500 3 0 .9703 .8574 .7290 .6141 .5120 .4219 .3430 .2963 .2746 .2160 .1644 .1327 .1250
1 .0294 .1354 .2430 .3251 .3840 .4219 .4410 .4444 .4436 .4320 .4084 .3823 .3750 2 .0003 .0071 .0270 .0574 .0960 .1406 .1890 .2222 .2389 .2880 .3341 .3674 .3750 3 .0000 .0001 .0010 .0034 .0080 .0156 .0270 .0370 .0429 .0640 .0911 .1176 .1250 4 0 .9606 .8145 .6561 .5220 .4096 .3164 .2401 .1975 .1785 .1296 .0915 .0677 .0625
1 .0388 .1715 .2916 .3685 .4096 .4219 .4116 .3951 .3845 .3456 .2995 .2600 .2500 2 .0006 .0135 .0486 .0975 .1536 .2109 .2646 .2963 .3105 .3456 .3675 .3747 .3750 3 .0000 .0005 .0036 .0115 .0256 .0469 .0756 .0988 .1115 .1536 .2005 .2400 .2500 4 .0000 .0000 .0001 .0005 .0016 .0039 .0081 .0123 .0150 .0256 .0410 .0576 .0625 5 0 .9510 .7738 .5905 .4437 .3277 .2373 .1681 .1317 .1160 .0778 .0503 .0345 .0312
1 .0480 .2036 .3280 .3915 .4096 .3955 .3602 .3292 .3124 .2592 .2059 .1657 .1562 2 .0010 .0214 .0729 .1582 .2043 .2637 .3087 .3292 .3364 .3456 .3369 .3185 .3125 3 .0000 .0011 .0081 .0244 .0512 .0879 .1323 .1646 .1811 .2304 .2757 .3060 .3125 4 .0000 .0000 .0004 .0022 .0064 .0145 .0284 .0412 .0488 .0768 .1128 .1470 .1562 5 .0000 .0000 .0000 .0001 .0003 .0010 .0024 .0041 .0053 .0102 .0185 .0283 .0312 6 0 .9415 .7351 .5314 .3771 .2621 .1780 .1176 .0872 .0754 .0467 .0277 .0176 .0156
1 .0571 .2321 .3543 .3993 .3932 .3560 .3025 .2634 .2437 .1866 .1359 .1014 .0938 2 .0014 .0305 .0984 .1762 .2458 .2966 .3241 .3292 .3280 .3110 .2780 .2437 .2344 3 .0000 .0021 .0146 .0415 .0819 .1312 .1852 .2195 .2355 .2765 .3032 .3121 .3125 4 .0000 .0001 .0012 .0055 .0154 .0330 .0592 .0823 .0951 .1382 .1861 .2249 .2344 5 .0000 .0000 .0001 .0004 .0015 .0044 .0102 .0165 .0205 .0369 .0609 .0864 .0938 6 .0000 .0000 .0000 .0000 .0002 .0002 .0007 .0014 .0018 .0041 .0083 .0139 .0156 7 0 .9321 .6983 .4783 .3206 .2097 .1335 .0824 .0585 .0490 .0280 .0152 .0090 .0078
1 .0659 .2575 .3720 .3960 .3670 .3115 .2471 .2048 .1848 .1306 .0872 .0603 .0547 2 .0020 .0406 .1240 .2097 .2753 .3115 .3177 .3073 .2985 .2613 .2140 .1740 .1641 3 .0000 .0036 .0230 .0617 .1147 .1730 .2269 .2561 .2679 .2903 .1918 .2786 .2734 4 .0000 .0002 .0026 .0109 :0287 .0577 .0972 .1280 .1442 .1935 .2388 .2676 .2734 5 .0000 .0000 .0002 .0012 .0043 .0115 .0250 .0384 .0466 .0774 .1172 .1543 .1641 6 .0000 .0000 .0000 .0001 .0004 .0013 .0036 .0064 .0084 .0172 .0320 .0494 .0547 7 .0000 .0000 .0000 .0000 .0000 .0001 .0002 .0005 .0006 .0016 .0037 .0068 .0078 8 0 .9227 .6634 .4305 .2725 .1678 .1001 .0576 .0390 .0319 .0168 .0084 .0046 .0039
1 .0746 .2793 .3826 .3847 .3355 .2670 .1977 .1373 .1373 .0896 .0548 .0352 .0312 2 .0001 .0515 .1488 .2376 .2936 .3115 .2965 .2587 .2587 .2090 .1569 .1183 .1 094 3 .0000 .0054 .0351 .0839 .1468 .2076 .2541 .2786 .2786 .2787 .2568 .2273 .2188 4 .0000 .0004 .0046 .0185 .0459 .0865 .1361 .1875 .1875 .2322 .2627 .2730 .2734 5 .0000 .0000 .0004 .0026 .0092 .0251 .0467 .0808 .0808 .1239 .1719 .2098 .2188 6 .0000 .0000 .0000 .0002 .0011 .0038 .0100 .0217 .0217 .0413 .0703 .1008 .1094 7 .0000 .0000 .0000 .0000 .0001 .0004 .0012 .0033 .0033 .0079 .0164 .0277 .0312
8 .0000 .0000 .0000 .0000 .0000 .0000 .0001 .0002 .0002 .0007 .0017 .0033 .0039
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Bioestadística FMCS Reus Curs 2013-14 URV 23
Exercici 5
X: Nombre de persones sense crisi de migranyaY: Nombre de persones amb crisi de migranyaY~B(8,0’1)
a) P(X≥7) = P(Y≤1) = P(Y=1)+P(Y=0) = 0’4305 + 0’3826 = 0’8131
b) P(X>3) = P(Y<5) = 1 – P(Y≥5) =
= 1 – ( P(Y=5) + P(Y=6) + P(Y=7) + P(Y=8) ) =
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Bioestadística FMCS Reus Curs 2013-14 URV 24
Probabilitats binomials exactes
P{X = x} =
x
npx (1 – p)n – x n = 1, 2, ...,10
n x \ p .01 .05 .10 .15 .20 .25 .30 1/3 .35 .40 .45 .49 .50 2 0 .9801 .9025 .8100 .7225 .6400 .5625 .4900 .4444 .4225 .3600 .3025 .2601 .2500
1 .0198 .9050 .1800 .2550 .3200 .3750 .4200 .4444 .4550 .4800 .4950 .4998 .5000 2 .0001 .0025 .0100 .0225 .0400 .0625 .0900 .1111 .1225 .1600 .2050 .2401 .2500 3 0 .9703 .8574 .7290 .6141 .5120 .4219 .3430 .2963 .2746 .2160 .1644 .1327 .1250
1 .0294 .1354 .2430 .3251 .3840 .4219 .4410 .4444 .4436 .4320 .4084 .3823 .3750 2 .0003 .0071 .0270 .0574 .0960 .1406 .1890 .2222 .2389 .2880 .3341 .3674 .3750 3 .0000 .0001 .0010 .0034 .0080 .0156 .0270 .0370 .0429 .0640 .0911 .1176 .1250 4 0 .9606 .8145 .6561 .5220 .4096 .3164 .2401 .1975 .1785 .1296 .0915 .0677 .0625
1 .0388 .1715 .2916 .3685 .4096 .4219 .4116 .3951 .3845 .3456 .2995 .2600 .2500 2 .0006 .0135 .0486 .0975 .1536 .2109 .2646 .2963 .3105 .3456 .3675 .3747 .3750 3 .0000 .0005 .0036 .0115 .0256 .0469 .0756 .0988 .1115 .1536 .2005 .2400 .2500 4 .0000 .0000 .0001 .0005 .0016 .0039 .0081 .0123 .0150 .0256 .0410 .0576 .0625 5 0 .9510 .7738 .5905 .4437 .3277 .2373 .1681 .1317 .1160 .0778 .0503 .0345 .0312
1 .0480 .2036 .3280 .3915 .4096 .3955 .3602 .3292 .3124 .2592 .2059 .1657 .1562 2 .0010 .0214 .0729 .1582 .2043 .2637 .3087 .3292 .3364 .3456 .3369 .3185 .3125 3 .0000 .0011 .0081 .0244 .0512 .0879 .1323 .1646 .1811 .2304 .2757 .3060 .3125 4 .0000 .0000 .0004 .0022 .0064 .0145 .0284 .0412 .0488 .0768 .1128 .1470 .1562 5 .0000 .0000 .0000 .0001 .0003 .0010 .0024 .0041 .0053 .0102 .0185 .0283 .0312 6 0 .9415 .7351 .5314 .3771 .2621 .1780 .1176 .0872 .0754 .0467 .0277 .0176 .0156
1 .0571 .2321 .3543 .3993 .3932 .3560 .3025 .2634 .2437 .1866 .1359 .1014 .0938 2 .0014 .0305 .0984 .1762 .2458 .2966 .3241 .3292 .3280 .3110 .2780 .2437 .2344 3 .0000 .0021 .0146 .0415 .0819 .1312 .1852 .2195 .2355 .2765 .3032 .3121 .3125 4 .0000 .0001 .0012 .0055 .0154 .0330 .0592 .0823 .0951 .1382 .1861 .2249 .2344 5 .0000 .0000 .0001 .0004 .0015 .0044 .0102 .0165 .0205 .0369 .0609 .0864 .0938 6 .0000 .0000 .0000 .0000 .0002 .0002 .0007 .0014 .0018 .0041 .0083 .0139 .0156 7 0 .9321 .6983 .4783 .3206 .2097 .1335 .0824 .0585 .0490 .0280 .0152 .0090 .0078
1 .0659 .2575 .3720 .3960 .3670 .3115 .2471 .2048 .1848 .1306 .0872 .0603 .0547 2 .0020 .0406 .1240 .2097 .2753 .3115 .3177 .3073 .2985 .2613 .2140 .1740 .1641 3 .0000 .0036 .0230 .0617 .1147 .1730 .2269 .2561 .2679 .2903 .1918 .2786 .2734 4 .0000 .0002 .0026 .0109 :0287 .0577 .0972 .1280 .1442 .1935 .2388 .2676 .2734 5 .0000 .0000 .0002 .0012 .0043 .0115 .0250 .0384 .0466 .0774 .1172 .1543 .1641 6 .0000 .0000 .0000 .0001 .0004 .0013 .0036 .0064 .0084 .0172 .0320 .0494 .0547 7 .0000 .0000 .0000 .0000 .0000 .0001 .0002 .0005 .0006 .0016 .0037 .0068 .0078 8 0 .9227 .6634 .4305 .2725 .1678 .1001 .0576 .0390 .0319 .0168 .0084 .0046 .0039
1 .0746 .2793 .3826 .3847 .3355 .2670 .1977 .1373 .1373 .0896 .0548 .0352 .0312 2 .0001 .0515 .1488 .2376 .2936 .3115 .2965 .2587 .2587 .2090 .1569 .1183 .1 094 3 .0000 .0054 .0351 .0839 .1468 .2076 .2541 .2786 .2786 .2787 .2568 .2273 .2188 4 .0000 .0004 .0046 .0185 .0459 .0865 .1361 .1875 .1875 .2322 .2627 .2730 .2734 5 .0000 .0000 .0004 .0026 .0092 .0251 .0467 .0808 .0808 .1239 .1719 .2098 .2188 6 .0000 .0000 .0000 .0002 .0011 .0038 .0100 .0217 .0217 .0413 .0703 .1008 .1094 7 .0000 .0000 .0000 .0000 .0001 .0004 .0012 .0033 .0033 .0079 .0164 .0277 .0312
8 .0000 .0000 .0000 .0000 .0000 .0000 .0001 .0002 .0002 .0007 .0017 .0033 .0039
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Bioestadística FMCS Reus Curs 2013-14 URV 25
Exercici 5
X: Nombre de persones sense crisi de migranyaY: Nombre de persones amb crisi de migranyaY~B(8,0’1)
a) P(X≥7) = P(Y≤1) = P(Y=1)+P(Y=0) = 0’4305 + 0’3826 = 0’8131
b) P(X>3) = P(Y<5) = 1 – P(Y≥5) =
= 1 – ( P(Y=5) + P(Y=6) + P(Y=7) + P(Y=8) ) =
= 1 – [ 0.0004 + 0 + 0 + 0] = 0.9996
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Bioestadística FMCS Reus Curs 2013-14 URV 26
El nombre de successos que ocurreixen en un interval de temps, de longitud, de espai segueix una distribució de Poisson si
La probabilitat de un succés es la mateixa en tot l’interval
La probabilitat de un succés no depèn dels successos ocorreguts amb anterioritat
Variable aleatòria de Poisson
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Bioestadística FMCS Reus Curs 2013-14 URV 27
Variable aleatòria de Poisson
X~P(λ)
La variable X de Poisson queda caracteritzada amb un únic paràmetre λ (Nombre mig de successos en un interval)
E(X)=λ V(X)=λ La seva funció de probabilitat és:
!x
e)xX(p
x -
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Bioestadística FMCS Reus Curs 2013-14 URV 28Bioestadística FMCS Reus Curs 2012-13 URV 28
Exercici 6
X: Nombre de casos en una hora
X~Poisson(λ=2)
24 hores 48 1 hora 2
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Bioestadística FMCS Reus Curs 2013-14 URV 29Bioestadística FMCS Reus Curs 2012-13 URV 29
Exercici 6
X~Poisson(λ=2)
λ \ x 0 1 2 3 4 5 6 7 8 9
2 0,13530,27070,27070,18040,09020,03610,01200,00340,00090,0002
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Bioestadística FMCS Reus Curs 2013-14 URV 3030
Exercici 6
b) P(X=0) = 0,1353
X: Nombre de casos en una horaX~Poisson(2)
c) P(X=3) = 0,1804
d) P(X≥5) = 1 - P(X<5) =
1 - (P(X=0)+P(X=1)+P(X=2)+P(X=3)+P(X=4)) =
1 - (0,1353+0,2707+0,2707+0,1804+0,0902) =
1 - 0,9473 = 0,0527
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Bioestadística FMCS Reus Curs 2013-14 URV 31
Exemple en que la variable aleatòria generada es pot considerar com variable de Poisson
Una població amb una prevalença de diabetis d'un 2%. La probabilitat de trobar 0, 1, 2, 3, 4,... diabètics en una mostra de 100 individus : B(100, 0’02)
En les mateixes condicions que la variable binomial si la característica és poc freqüent (p<0.05)
mostres grans (n≥30)
La variable X segueix el model de distribució de Poisson
Binomial (100,0’02) Poisson (2)
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Bioestadística FMCS Reus Curs 2013-14 URV 32
Exercici 7
N=1200 P=0’001 X: Nombre de persones amb reacció al·lèrgica X~B(1200,0’001)
X~B(1200,0’001)
N gran
X~Poisson(λ=1200·0’001)
X~Poisson(λ=1’2)
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Bioestadística FMCS Reus Curs 2013-14 URV 33
Exercici 7
X~Poisson(λ=1’2)
b) p(X=2) = 0’2169
c) p(X>2) = 1 – [ p(X=0) + p(X=1) + p(X=2) ] =
= 1 – [ 0’3012 + 0’3614 + 0’2169] =
= 1 – 0’8795 = 0’1205
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Bioestadística FMCS Reus Curs 2013-14 URV 34
Exercici 8
A cada interval X: Nombre de particules emeses X~Poisson(λ=4)
a) P(X≥4) = 1 – [ p(X=0) + p (X=1) + p (X=2) + p(X=3) ] =
= 1 – [0’0183 + 0’0733 + 0’1465 + 0’1954] =
= 1 – 0’4335 = 0’5665
1 segon 0’5 8 segons 4
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Bioestadística FMCS Reus Curs 2013-14 URV 35
Exercici 8
En 5 intervals Y: Nombre de intervals que emeten 4 o mes particules Y~B(5,0’5665)
b) P(Y≥1) = 1 – [ p(Y=0)] =
= 1 – [0,0153] = 0’9847
( ) 5050 )4335'0(5665'0=)0=Y(p
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Bioestadística FMCS Reus Curs 2013-14 URV 36
Variable aleatoria normal
P(a≤X≤b)=P(X≥a) – P(X≥b)
P(Z>-a) = P(Z<a)
P(Z<a) = 1 - P(Z>a)
P(Z>-a) = 1 - P(Z>a)
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Bioestadística FMCS Reus Curs 2013-14 URV 37
Exercici 9
El pes de les persones d'una determinada població es distribueix normalment amb una mitjana de 80 kg. i una desviació típica 10 kg.
a. Quina és la probabilitat de que una persona pesi entre 70 i 90 kg?
• Quina és la probabilitat de que una persona pesi més de 95 Kg
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Bioestadística FMCS Reus Curs 2013-14 URV 38
Exercici 9
El pes de les persones d'una determinada població es distribueix normalment amb una mitjana de 80 kg. i una desviació típica 10 kg.
P(70 < X < 90) = P (X > 70) – P (X > 90)
X~N(80,10) Z~N(0,1)10
X=
σμX
=Z80--
a a - 80
10
P(X>a) P(Z > (a-80) / 10 )
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Bioestadística FMCS Reus Curs 2013-14 URV 39
Exercici 9
70 8010 X 80
10
X: N(80,10)
P(70 < X < 90) = P (X > 70) – P (X > 90) =
= P ( > ) - P ( > )
= P (Z > -1) – P (Z > 1) =
= [1 – P(Z>1)] – P(Z>1) =
= 1 – 2·P(Z>1) =
X 8010 90 80
10
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Bioestadística FMCS Reus Curs 2013-14 URV 40
P(z) 0,00 0,01 0,02 0,03 0,04 0,05 0,06 0,07 0,08 0,09
0,0 0,5000 0,4960 0,4920 0,4880 0,4841 0,4801 0,4761 0,4721 0,4681 0,4641
0,1 0,4602 0,4562 0,4522 0,4483 0,4443 0,4404 0,4364 0,4325 0,4286 0,4247
0,2 0,4207 0,4168 0,4129 0,4091 0,4052 0,4013 0,3974 0,3936 0,3897 0,3859
0,3 0,3821 0,3783 0,3745 0,3707 0,3669 0,3632 0,3594 0,3557 0,3520 0,3483
0,4 0,3446 0,3409 0,3372 0,3336 0,3300 0,3264 0,3228 0,3192 0,3156 0,3121
0,5 0,3085 0,3050 0,3015 0,2981 0,2946 0,2912 0,2877 0,2843 0,2810 0,2776
0,6 0,2743 0,2709 0,2676 0,2644 0,2611 0,2579 0,2546 0,2514 0,2483 0,2451
0,7 0,2420 0,2389 0,2358 0,2327 0,2297 0,2266 0,2236 0,2207 0,2177 0,2148
0,8 0,2119 0,2090 0,2061 0,2033 0,2005 0,1977 0,1949 0,1922 0,1894 0,1867
0,9 0,1841 0,1814 0,1788 0,1762 0,1736 0,1711 0,1685 0,1660 0,1635 0,1611
1,0 0,1587 0,1563 0,1539 0,1515 0,1492 0,1469 0,1446 0,1423 0,1401 0,1379
1,1 0,1357 0,1335 0,1314 0,1292 0,1271 0,1251 0,1230 0,1210 0,1190 0,1170
1,2 0,1151 0,1131 0,1112 0,1094 0,1075 0,1057 0,1038 0,1020 0,1003 0,0985
1,3 0,0968 0,0951 0,0934 0,0918 0,0901 0,0885 0,0869 0,0853 0,0838 0,0823
1,4 0,0808 0,0793 0,0778 0,0764 0,0749 0,0735 0,0721 0,0708 0,0694 0,0681
1,5 0,0668 0,0655 0,0643 0,0630 0,0618 0,0606 0,0594 0,0582 0,0571 0,0559
1,6 0,0548 0,0537 0,0526 0,0516 0,0505 0,0495 0,0485 0,0475 0,0465 0,0455
1,7 0,0446 0,0436 0,0427 0,0418 0,0409 0,0401 0,0392 0,0384 0,0375 0,0367
1,8 0,0359 0,0352 0,0344 0,0336 0,0329 0,0322 0,0314 0,0307 0,0301 0,0294
1,9 0,0287 0,0281 0,0274 0,0268 0,0262 0,0256 0,0250 0,0244 0,0239 0,0233
Distribució normal (0;1) P ( X ≥ a ) a
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Bioestadística FMCS Reus Curs 2013-14 URV 41
Exercici 9
70 8010 X 80
10
X: N(80,10)
P(70 < X < 90) = P (X > 70) – P (X > 90) =
= P ( > ) - P ( > )
= P (Z > -1) – P (Z > 1) =
= [1 – P(Z>1)] – P(Z>1) =
= 1 – 2·P(Z>1) = 1 – 2·0’1587 = 0’6826
X 8010 90 80
10
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Bioestadística FMCS Reus Curs 2013-14 URV 42
Exercici 9
10
80X10
8095
X: N(80,10)
P (X > 95) =
= P ( > )
= P (Z > 1’5) =
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Bioestadística FMCS Reus Curs 2013-14 URV 43
P(z) 0,00 0,01 0,02 0,03 0,04 0,05 0,06 0,07 0,08 0,09
0,0 0,5000 0,4960 0,4920 0,4880 0,4841 0,4801 0,4761 0,4721 0,4681 0,4641
0,1 0,4602 0,4562 0,4522 0,4483 0,4443 0,4404 0,4364 0,4325 0,4286 0,4247
0,2 0,4207 0,4168 0,4129 0,4091 0,4052 0,4013 0,3974 0,3936 0,3897 0,3859
0,3 0,3821 0,3783 0,3745 0,3707 0,3669 0,3632 0,3594 0,3557 0,3520 0,3483
0,4 0,3446 0,3409 0,3372 0,3336 0,3300 0,3264 0,3228 0,3192 0,3156 0,3121
0,5 0,3085 0,3050 0,3015 0,2981 0,2946 0,2912 0,2877 0,2843 0,2810 0,2776
0,6 0,2743 0,2709 0,2676 0,2644 0,2611 0,2579 0,2546 0,2514 0,2483 0,2451
0,7 0,2420 0,2389 0,2358 0,2327 0,2297 0,2266 0,2236 0,2207 0,2177 0,2148
0,8 0,2119 0,2090 0,2061 0,2033 0,2005 0,1977 0,1949 0,1922 0,1894 0,1867
0,9 0,1841 0,1814 0,1788 0,1762 0,1736 0,1711 0,1685 0,1660 0,1635 0,1611
1,0 0,1587 0,1563 0,1539 0,1515 0,1492 0,1469 0,1446 0,1423 0,1401 0,1379
1,1 0,1357 0,1335 0,1314 0,1292 0,1271 0,1251 0,1230 0,1210 0,1190 0,1170
1,2 0,1151 0,1131 0,1112 0,1094 0,1075 0,1057 0,1038 0,1020 0,1003 0,0985
1,3 0,0968 0,0951 0,0934 0,0918 0,0901 0,0885 0,0869 0,0853 0,0838 0,0823
1,4 0,0808 0,0793 0,0778 0,0764 0,0749 0,0735 0,0721 0,0708 0,0694 0,0681
1,5 0,0668 0,0655 0,0643 0,0630 0,0618 0,0606 0,0594 0,0582 0,0571 0,0559
1,6 0,0548 0,0537 0,0526 0,0516 0,0505 0,0495 0,0485 0,0475 0,0465 0,0455
1,7 0,0446 0,0436 0,0427 0,0418 0,0409 0,0401 0,0392 0,0384 0,0375 0,0367
1,8 0,0359 0,0352 0,0344 0,0336 0,0329 0,0322 0,0314 0,0307 0,0301 0,0294
1,9 0,0287 0,0281 0,0274 0,0268 0,0262 0,0256 0,0250 0,0244 0,0239 0,0233
Distribució normal (0;1) P ( X ≥ a ) a
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Bioestadística FMCS Reus Curs 2013-14 URV 44
Exercici 9
10
80X10
8095
X: N(80,10)
P (X > 95) =
= P ( > )
= P (Z > 1’5) = 0’0668
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Bioestadística FMCS Reus Curs 2013-14 URV 45
Exercici 10
hipertens "borderline" : Entre 90 i 100 mmHg inclusius. Les pressions sanguínies diastòliques dels quals presenten una distribució normal amb una mitjana de 80 i variança 144 (σ=12)
P(90 < X < 100) = P (X > 90) – P (X > 100)
X~N(80,12) Z~N(0,1)12X
=σ
μX=Z
80--
a a - 80
12
P(X>a) P(Z > (a-80) / 12 )
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Bioestadística FMCS Reus Curs 2013-14 URV 46
Exercici 10
90 8012 X 80
12
X: N(80,12)
P(90 < X < 100) = P (X > 90) – P (X > 100) =
= P ( > ) - P ( > )
= P (Z > 0’83) – P (Z > 1’67) =
X 8012 100 80
12
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Bioestadística FMCS Reus Curs 2013-14 URV 47
P(z) 0,00 0,01 0,02 0,03 0,04 0,05 0,06 0,07 0,08 0,09
0,0 0,5000 0,4960 0,4920 0,4880 0,4841 0,4801 0,4761 0,4721 0,4681 0,4641
0,1 0,4602 0,4562 0,4522 0,4483 0,4443 0,4404 0,4364 0,4325 0,4286 0,4247
0,2 0,4207 0,4168 0,4129 0,4091 0,4052 0,4013 0,3974 0,3936 0,3897 0,3859
0,3 0,3821 0,3783 0,3745 0,3707 0,3669 0,3632 0,3594 0,3557 0,3520 0,3483
0,4 0,3446 0,3409 0,3372 0,3336 0,3300 0,3264 0,3228 0,3192 0,3156 0,3121
0,5 0,3085 0,3050 0,3015 0,2981 0,2946 0,2912 0,2877 0,2843 0,2810 0,2776
0,6 0,2743 0,2709 0,2676 0,2644 0,2611 0,2579 0,2546 0,2514 0,2483 0,2451
0,7 0,2420 0,2389 0,2358 0,2327 0,2297 0,2266 0,2236 0,2207 0,2177 0,2148
0,8 0,2119 0,2090 0,2061 0,2033 0,2005 0,1977 0,1949 0,1922 0,1894 0,1867
0,9 0,1841 0,1814 0,1788 0,1762 0,1736 0,1711 0,1685 0,1660 0,1635 0,1611
1,0 0,1587 0,1563 0,1539 0,1515 0,1492 0,1469 0,1446 0,1423 0,1401 0,1379
1,1 0,1357 0,1335 0,1314 0,1292 0,1271 0,1251 0,1230 0,1210 0,1190 0,1170
1,2 0,1151 0,1131 0,1112 0,1094 0,1075 0,1057 0,1038 0,1020 0,1003 0,0985
1,3 0,0968 0,0951 0,0934 0,0918 0,0901 0,0885 0,0869 0,0853 0,0838 0,0823
1,4 0,0808 0,0793 0,0778 0,0764 0,0749 0,0735 0,0721 0,0708 0,0694 0,0681
1,5 0,0668 0,0655 0,0643 0,0630 0,0618 0,0606 0,0594 0,0582 0,0571 0,0559
1,6 0,0548 0,0537 0,0526 0,0516 0,0505 0,0495 0,0485 0,0475 0,0465 0,0455
1,7 0,0446 0,0436 0,0427 0,0418 0,0409 0,0401 0,0392 0,0384 0,0375 0,0367
1,8 0,0359 0,0352 0,0344 0,0336 0,0329 0,0322 0,0314 0,0307 0,0301 0,0294
1,9 0,0287 0,0281 0,0274 0,0268 0,0262 0,0256 0,0250 0,0244 0,0239 0,0233
Distribució normal (0;1) P ( X ≥ a ) a
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Bioestadística FMCS Reus Curs 2013-14 URV 48
Exercici 10
90 8012 X 80
12
X: N(80,12)
P(90 < X < 100) = P (X > 90) – P (X > 100) =
= P ( > ) - P ( > )
= P (Z > 0’83) – P (Z > 1’67) =
= 0’2033 – 0’0475 = 0,1558
X 8012 100 80
12
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Bioestadística FMCS Reus Curs 2013-14 URV 49
Exercici 10
12
80X12
80104
X: N(80,12)
P (X > 104) =
= P ( > )
= P (Z > 2’0) =
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Curs 2013-14 URV 50
P(z) 0,00 0,01 0,02 0,03 0,04 0,05 0,06 0,07 0,08 0,09
0,0 0,5000 0,4960 0,4920 0,4880 0,4841 0,4801 0,4761 0,4721 0,4681 0,4641
0,1 0,4602 0,4562 0,4522 0,4483 0,4443 0,4404 0,4364 0,4325 0,4286 0,4247
0,2 0,4207 0,4168 0,4129 0,4091 0,4052 0,4013 0,3974 0,3936 0,3897 0,3859
0,3 0,3821 0,3783 0,3745 0,3707 0,3669 0,3632 0,3594 0,3557 0,3520 0,3483
0,4 0,3446 0,3409 0,3372 0,3336 0,3300 0,3264 0,3228 0,3192 0,3156 0,3121
0,5 0,3085 0,3050 0,3015 0,2981 0,2946 0,2912 0,2877 0,2843 0,2810 0,2776
0,6 0,2743 0,2709 0,2676 0,2644 0,2611 0,2579 0,2546 0,2514 0,2483 0,2451
0,7 0,2420 0,2389 0,2358 0,2327 0,2297 0,2266 0,2236 0,2207 0,2177 0,2148
0,8 0,2119 0,2090 0,2061 0,2033 0,2005 0,1977 0,1949 0,1922 0,1894 0,1867
0,9 0,1841 0,1814 0,1788 0,1762 0,1736 0,1711 0,1685 0,1660 0,1635 0,1611
1,0 0,1587 0,1563 0,1539 0,1515 0,1492 0,1469 0,1446 0,1423 0,1401 0,1379
1,1 0,1357 0,1335 0,1314 0,1292 0,1271 0,1251 0,1230 0,1210 0,1190 0,1170
1,2 0,1151 0,1131 0,1112 0,1094 0,1075 0,1057 0,1038 0,1020 0,1003 0,0985
1,3 0,0968 0,0951 0,0934 0,0918 0,0901 0,0885 0,0869 0,0853 0,0838 0,0823
1,4 0,0808 0,0793 0,0778 0,0764 0,0749 0,0735 0,0721 0,0708 0,0694 0,0681
1,5 0,0668 0,0655 0,0643 0,0630 0,0618 0,0606 0,0594 0,0582 0,0571 0,0559
1,6 0,0548 0,0537 0,0526 0,0516 0,0505 0,0495 0,0485 0,0475 0,0465 0,0455
1,7 0,0446 0,0436 0,0427 0,0418 0,0409 0,0401 0,0392 0,0384 0,0375 0,0367
1,8 0,0359 0,0352 0,0344 0,0336 0,0329 0,0322 0,0314 0,0307 0,0301 0,0294
1,9 0,0287 0,0281 0,0274 0,0268 0,0262 0,0256 0,0250 0,0244 0,0239 0,0233
Distribució normal (0;1) P ( X ≥ a ) a
2,0 0,0228
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Bioestadística FMCS Reus Curs 2013-14 URV 51
Exercici 10
12
80X12
80104
X: N(80,12)
P (X > 104) =
= P ( > )
= P (Z > 2’0) = 0’0228