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Probabilidades Discretas: Modelos Binomial y Poisson, Ejercicios de Biometría

Universidad de Granada (UGR)Biometría

Documento que presenta el cálculo de probabilidades discretas a través del modelo binomial y poisson. Contiene ejemplos con distintas variables y parámetros.

Tipo: Ejercicios

2020/2021

Subido el 29/09/2022

1303natalia 🇪🇸

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Modelos de Probabilidad

Modelo Binomial

Gráficamente P(D) = 0,10

x  B (n=5, P=0,10)

n = 0 1 2 3 4 5 n= 5

P= 0.1

Funcionamiento de los cáculos en Excel:

nº de éxitos = k P (X=k) P (X=k)

P(X=2) 2 0 P (X=0) 0.59049 0.59049 P(X=2) =

1 P (X=1) 0.32805 0.91854

P(X≤3) 0 1 2 3 2 P (X=2) 0.07290 0.99144 P(X≤3) =

3 P (X=3) 0.00810 0.99954

P(X>2) = 1-P(X≤2) 3 4 5 = 0 1 2 4 P (X=4) 0.00045 0.99999 P(X>2) = 1-P(X≤2)=

5 P (X=5) 0.00001 1.00000

P(3≤X≤5) 3 4 5 ∑ 1.00000 P(3≤X≤5) =

ó P(3≤X≤5) =

P(X=0) 0 ó P(3≤X≤5) =

P(X=0) =

Falso: Probabilidad puntual = Función de probabilidad bruta = 0

Verdadero: Función de probabilidad acumulada = 1

Probabilidad

(Falso)

Acumulada

(Verdadero)

Descubre Ejercicios de Biometría Universidad de Granada (UGR)

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¡Descarga Probabilidades Discretas: Modelos Binomial y Poisson y más Ejercicios en PDF de Biometría solo en Docsity!

Modelos de Probabilidad

Modelo Binomial

Gráficamente P(D) = 0,10 x͠ B (n=5, P=0,10)

n = 0 1 2 3 4 5 n= 5 P= 0.1 Funcionamiento d nº de éxitos = k P (X=k) P (X=k)

P(X=2) 2 0 P (X=0) 0.59049 0.59049 P(X=2) = 1 P (X=1) 0.32805 0. P(X≤3) 0 1 2 3 2 P (X=2) 0.07290 0.99144 P(X≤3) = 3 P (X=3) 0.00810 0. P(X>2) = 1-P(X≤2) 3 4 5 = 0 1 2 4 P (X=4) 0.00045 0.99999 P(X>2) = 1-P(X≤2)= 5 P (X=5) 0.00001 1. P(3≤X≤5) 3 4 5 ∑ 1.00000 P(3≤X≤5) = ó P(3≤X≤5) = P(X=0) 0 ó P(3≤X≤5) =

P(X=0) =

Falso: Probabilidad puntual = Función de probabilidad bruta = 0 Verdadero: Función de probabilidad acumulada = 1

Probabilidad (Falso)

Acumulada (Verdadero)

de los cáculos en Excel: Resultado Posición/Contenido de la celda

0.07290 = R

0.99954 = P(X=0) + P(X=1) + P(X=2) + P(X=3)

ó = (Acumulada x=2) = S 0.00856 = 1 - (Acumulada x=2) = 1 - S 0.00856 = P(X=3) + P(X=4) + P(X=5) 0.00856 = (Acumulada de x=5) - ((Acumulada de x=3)-una posición) = S13-S 0.00856 = DISTR.BINOM(P13;Q5;Q6;VERDADERO) - DISTR.BINOM(P11-1;Q5;Q6;VERDADERO)

0.59049 = R

DISTR.BINOM(C11;G11;H11;FALSO)

DISTR.BINOM(C12;G12;H12;I12)+DISTR.BINOM(D12;G12;H12;I12)+DISTR.BINOM(E12;G12;H12;I12)+DISTR.BINOM(F12;G12;H12;I12)

DISTR.BINOM(C13;G13;H13;I13)

1-DISTR.BINOM(C14;F14;G14;VERDADERO)

DISTR.BINOM(C15;G15;H15;I15)+DISTR.BINOM(D15;G15;H15;I15)+DISTR.BINOM(E15;G15;H15;I15)

DISTR.BINOM(C16;G16;H16;I16)

Descripción de la celda

P(x=k)=DISTR.BINOM(múm-éxito;ensayos;prob_éxito;prob puntual o acumulada)

Modelos de Probabilidad Discretos - Poisson

Ej.2) x: nº de suicidios al mes

λ = 2,60 (Promedio de suicidios al mes)

1 mes 2.

2 meses 5.

3 meses 7.

Modelo Poisson λ k P(X=k)

a) 2.6 3 1 0.

b) p(2≤x≤4)^ 2.6^2 3 4 0 0.

*b) p(x≤4)-P(x≤1)^ 2.6^4 1 1 0.

c) P(x=0) 5.2 0 0 0.

d) P(x<3)= P(x≤2) 7.8 2 1 0.

media = λ x en Excel es k

FALSO: Probabilidad puntual = Función de probabilidad bruta = 0

VERDADERO: Función de probabilidad acumulada = 1

*b) Otra forma de resolver b, tomando en consideración la probabilidad acumulada

Puntual/

Acumulada

p(x>3)= 1-p(x≤3)

Modelos de Probabilidad Continuos - Dist. Normal

Ej.3) medida antropométrica de la población (medido IMC)

Delgados ≤ 20 Obesos > 30

Modelo Normal k μ σ P(X=k) DISTR.NORM(x;μ;σ;VERDADERO)

a) 20 25.2 2.8 1 0.03165 DISTR.NORM(C11;E11;F11;VERDADERO)

b) 30 25.2 2.8 1 0.04324 1-DISTR.NORM(C12;E12;F12;VERDADERO) c) P(22≤x≤28) 22 28 25.2 2.8 1 0.71480 DISTR.NORM(D13;E13;F13;G13)-DISTR.NORM(C13;E13;F13;G13)

d) Percentiles en la Distibución normal Percentil

5% 0.05 25.2 2.8 20.59 DISTR.NORM.INV(C16;E16;F16) 95% 0.95 25.2 2.8 29.81 DISTR.NORM.INV(C17;E17;F17)

x͠ N (μ=25,2 ; σ=2,8)

Puntual/

Acumulada

P(X≤20)

P(X>30) = 1- P(X≤30)

Hombres Mujeres

DISTRIBUCION NORMAL

176.65 72.69 150.50 81.56 0. Estaturas Pesos Estaturas Pesos
176.19 52.97 162.38 77.00 0.
168.13 59.92 164.94 61.69 0.
176.65 61.22 154.26 65.94 0.
172.78 83.33 164.36 53.63 0.
152.08 70.07 163.79 76.42 0.
165.12 77.59 153.23 60.17 0.
144.86 70.73 160.45 62.00 0.
167.38 67.44 165.24 72.17 0.
162.42 88.67 161.00 75.86 0.
155.92 65.74 164.36 40.33 0.
155.42 51.21 157.64 56.26 0.
169.65 77.98 186.59 61.39 0.
140.74 71.05 152.87 59.86 0.
146.89 65.74 162.10 46.81 0.
164.74 65.04 158.50 59.55 0.
165.87 73.36 160.72 69.54 0.
177.12 72.36 147.51 52.80 0.
181.31 67.78 148.61 59.23 0.
181.31 74.71 150.93 65.94 0.
154.90 71.05 151.74 65.02 0.
161.22 65.39 175.10 52.80 0.
159.56 51.21 140.31 50.97 0.
174.44 80.44 157.64 61.09 0.
167.00 45.95 162.66 65.94 0.
155.42 53.74 166.14 65.94 0.
156.88 80.88 157.35 85.47 0.
148.60 65.04 169.13 82.64 0.
190.32 79.17 162.10 55.55 0.
193.26 68.11 162.66 59.23 0.
171.58 70.73 157.35 56.26 0.
166.25 61.22 161.55 63.50 0.
160.40 67.78 165.24 66.25 0.
180.17 60.79 150.06 38.26 0.
158.26 45.95 148.61 63.50 0.
190.32 71.05 170.26 74.35 0.
152.69 74.03 148.61 77.62 0.
187.11 71.38 181.69 65.94 0.
171.19 81.34 153.58 58.28 0.
195.21 50.19 162.94 65.32 0.
175.30 81.81 183.59 57.62 0.
201.90 77.59 160.45 54.81 0.
152.08 61.22 164.65 55.91 0.
158.26 55.13 178.10 61.09 0.
171.98 100.64 181.69 66.57 0.
157.81 68.44 149.60 55.91 0.
184.62 54.46 176.46 61.09 0.
147.78 54.46 166.77 56.26 0.
184.62 59.92 171.94 50.46 0.
155.92 52.97 158.50 56.95 0.
191.67 52.97 150.93 57.29 0.
162.81 76.48 173.39 49.94 0.
143.68 59.46 161.55 78.28 0.
142.33 56.36 159.90 74.83 0.
163.59 63.95 159.06 65.32 0.
165.49 52.97 170.26 38.26 0.
153.83 65.39 164.65 58.28 0.
180.73 83.33 151.74 48.80 0.
174.02 72.69 155.55 87.54 0.
175.30 49.02 161.83 46.81 0.
187.11 81.81 155.55 61.39 0.
153.83 70.40 153.58 62.90 0.
174.87 88.67 164.65 75.34 0.
175.30 77.22 143.90 62.90 0.
174.44 59.92 157.64 56.26 0.
175.30 73.02 152.87 64.11 0.
163.59 76.85 158.50 49.38 0.
153.83 62.43 170.26 65.63 0.
144.86 80.01 180.26 78.99 0.
155.92 68.77 166.14 64.11 0.
176.19 65.74 169.49 57.62 0.
181.92 63.95 160.72 59.55 0.
174.44 67.78 161.83 73.00 0.
154.38 73.69 183.59 58.92 0.
182.55 59.92 152.13 53.22 0.
166.62 76.48 172.88 55.91 0.
179.10 77.22 174.49 49.38 0.
168.51 74.03 162.94 65.32 0.
190.32 62.82 183.59 62.60 0.
155.42 58.99 160.45 66.88 0.
173.60 76.48 179.09 34.98 0.
151.45 63.21 160.45 57.29 0.
171.98 45.95 150.06 63.20 0.
138.79 82.30 141.74 54.42 0.
177.59 71.38 169.13 46.81 0.
183.20 52.97 169.49 70.62 0.
175.30 82.30 158.78 50.97 0.
159.56 58.50 176.46 43.16 0.
163.20 87.06 156.16 59.86 0.
151.45 85.67 167.09 83.91 0.
171.19 77.59 181.69 61.39 0.
182.55 72.36 163.50 55.55 0.
143.68 82.80 177.23 44.24 0.
142.33 85.04 145.54 64.71 0.
162.42 67.44 158.78 68.85 0.
163.20 60.36 160.45 73.43 0.
132.10 79.58 157.64 55.91 0.
158.26 69.75 156.76 73.88 0.
181.31 69.42 169.87 66.57 0.
179.62 85.04 166.14 66.88 0.
174.02 52.13 170.26 63.50 0.
175.74 81.81 175.10 54.81 0.
146.89 74.03 186.59 51.45 0.
164.74 90.61 156.46 44.24 0.
155.92 65.74 163.22 76.42 0.
179.10 60.79 164.65 58.28 0.
160.40 55.76 170.26 69.54 0.
170.80 100.64 172.88 72.58 0.
153.83 62.43 179.09 62.00 0.
178.08 81.34 162.94 54.81 0.
167.75 90.61 164.94 61.09 0.
158.70 59.46 165.54 70.99 0.
175.30 68.11 159.62 69.54 0.
140.74 52.97 170.66 75.86 0.
173.19 69.09 159.06 59.55 0.
167.00 76.12 152.51 43.16 0.
161.62 49.02 156.16 57.29 0.
155.92 75.06 150.50 59.86 0.
146.89 61.22 155.86 60.78 0.
182.55 77.59 149.12 44.24 0.
195.21 82.30 154.91 62.60 0.
177.59 59.46 186.59 54.81 0.
142.33 82.30 153.23 48.80 0.
152.08 52.13 157.64 63.80 0.
177.59 80.01 153.92 52.80 0.
156.88 87.06 170.66 59.86 0.
183.20 70.73 164.94 68.51 0.
181.31 69.42 160.45 53.63 0.
163.20 69.75 147.51 74.83 0.
181.92 61.22 155.23 79.76 0.
169.26 73.69 143.90 67.20 0.
153.83 56.93 158.78 41.89 0.
150.80 79.58 160.72 80.61 0.
168.13 77.59 168.77 70.99 0.
178.08 73.02 152.87 53.22 0.
148.60 49.02 145.54 71.38 0.
170.41 74.37 168.42 54.81 0.
151.45 74.71 163.50 80.61 0.
177.59 58.99 176.46 90.82 0.
171.19 52.13 173.92 38.26 0.
174.44 47.64 163.22 55.91 0.
170.80 87.06 150.93 60.17 0.
191.67 87.83 142.91 45.19 0.
153.83 51.21 176.46 69.19 0.
195.21 79.58 164.94 47.52 0.
191.67 58.00 161.83 87.54 0.
190.32 70.73 157.06 58.60 0.
178.58 77.22 151.74 50.46 0.
179.10 71.38 156.16 52.80 0.
162.81 63.58 159.06 83.91 0.
147.78 87.06 148.61 75.34 0.
154.90 55.13 154.59 52.80 0.
175.30 81.34 158.21 61.09 0.
156.41 68.44 160.72 67.20 0.
170.80 61.22 156.16 54.42 0.
163.97 51.21 170.26 78.28 0.
177.12 63.58 146.25 52.80 0.
163.97 93.16 181.69 57.95 0.
150.80 55.13 158.78 72.17 0.
181.31 85.67 159.06 64.11 0.
151.45 59.92 147.51 68.18 0.
142.33 79.17 142.91 56.26 0.
184.62 61.22 162.38 61.39 0.
201.90 86.34 174.49 50.97 0.
166.62 76.48 155.23 83.91 0.
176.65 72.03 149.12 60.48 0.
162.42 83.87 159.06 68.18 0.
149.38 75.76 159.90 49.94 0.
183.20 70.40 160.72 34.98 0.
153.27 68.11 153.58 50.97 0.
167.38 76.85 152.51 75.86 0.
158.26 79.58 167.74 67.52 0.
145.92 74.03 166.45 65.63 0.
172.78 45.95 155.86 47.52 0.
158.70 79.17 163.50 70.62 0.
152.08 84.44 176.46 57.62 0.
184.62 80.01 153.92 81.56 0.
160.40 63.58 157.93 83.91 0.
197.81 63.21 143.90 56.61 0.
178.08 82.30 152.13 68.85 0.
179.62 79.17 174.49 56.61 0.
168.88 60.36 168.42 57.29 0.
162.81 62.43 147.51 80.61 0.
168.51 73.69 162.38 66.88 0.
161.22 51.21 161.00 54.42 0.
164.35 57.47 167.09 44.24 0.
184.62 74.37 164.65 49.38 0.
149.38 70.73 183.59 85.47 0.
173.19 97.10 162.66 59.86 0.
148.60 74.03 145.54 41.89 0.
153.83 74.71 150.93 71.38 0.
170.41 56.93 183.59 85.47 0.
144.86 59.46 180.26 71.77 0.
175.74 66.43 140.31 62.30 0.
136.19 77.22 164.65 57.29 0.
180.17 79.17 154.91 72.17 0.
164.74 80.01 154.26 72.17 0.
146.89 62.82 166.45 55.55 0.
140.74 53.74 165.24 70.25 0.
132.10 68.77 169.87 38.26 0.
197.81 82.80 173.39 64.71 0.
170.80 53.74 166.45 40.33 0.
149.38 83.33 162.66 73.43 0.
157.35 94.85 167.41 46.81 0.
189.14 52.97 159.62 43.16 0.
159.56 69.75 164.94 48.18 0.
167.00 84.44 152.87 78.28 0.
136.19 61.63 147.51 51.92 0.
151.45 70.73 147.51 64.71 0.
151.45 63.21 165.84 49.38 0.
157.81 89.59 162.38 64.41 0.
178.08 59.46 138.41 75.34 0.
168.88 87.83 152.87 65.63 0.
190.32 77.98 161.00 46.04 0.
174.44 69.42 141.74 43.16 0.
173.19 59.46 166.14 43.16 0.
174.02 51.21 162.10 78.28 0.
178.08 69.09 166.14 64.11 0.
180.73 68.77 163.79 54.81 0.
152.08 75.76 159.06 73.00 0.
150.80 89.59 157.35 61.39 0.
145.92 67.11 145.54 63.20 0.
153.83 76.85 175.10 53.63 0.
162.42 50.19 155.23 58.92 0.
153.27 93.16 171.07 80.61 0.
140.74 55.76 162.10 53.63 0.
166.62 65.04 170.66 45.19 0.
161.22 47.64 148.08 63.20 0.
173.19 45.95 160.45 60.78 0.
159.13 94.85 160.17 62.00 0.
179.10 64.68 160.72 61.69 0.
177.59 78.37 171.07 62.00 0.
146.89 97.10 152.87 63.50 0.
161.62 70.07 144.77 67.52 0.
164.35 70.40 149.12 65.63 0.
166.25 65.74 157.93 71.38 0.
195.21 71.71 169.87 73.88 0.
162.42 47.64 149.60 61.09 0.
151.45 75.76 156.76 74.35 0.
165.49 97.10 157.93 64.71 0.
132.10 93.16 164.65 60.17 0.
190.32 94.85 172.40 59.55 0.
136.19 58.99 169.49 69.54 0.
149.38 66.43 163.79 70.99 0.
162.81 84.44 179.09 62.00 0.
146.89 80.88 158.78 55.18 0.
176.65 69.75 183.59 70.25 0.
155.92 78.76 173.92 64.71 0.
172.78 74.03 176.46 49.94 0.
170.03 52.97 165.24 64.41 0.
201.90 56.36 169.13 61.69 0.
153.83 57.47 161.00 62.30 0.
186.22 66.43 161.55 51.92 0.
161.62 87.83 162.94 58.92 0.
158.26 70.07 176.46 46.81 0.
179.62 64.32 151.74 69.19 0.
174.87 49.02 148.08 73.43 0.
171.19 56.93 163.50 67.85 0.
162.02 91.78 176.46 55.55 0.
160.40 50.19 167.74 73.88 0.
176.65 73.02 159.90 63.50 0.
163.59 66.77 157.64 70.99 0.
176.65 68.77 170.26 80.61 0.
176.19 89.59 153.92 74.35 0.
156.88 60.79 164.94 62.30 0.
157.35 81.81 150.93 51.92 0.
152.08 70.40 150.50 66.57 0.
169.26 89.59 167.09 55.18 0.
168.51 78.37 168.42 53.63 0.
179.10 70.07 163.22 53.22 0.
156.41 75.06 161.28 71.38 0.
176.19 76.12 172.88 40.33 0.
155.42 54.46 171.94 55.91 0.
189.14 51.21 152.87 65.32 0.
177.59 68.77 161.00 54.03 0.
175.30 52.13 173.92 61.09 0.
169.26 68.77 178.10 62.90 0.
171.58 72.36 169.87 65.32 0.
148.60 87.06 157.06 38.26 0.
157.35 59.46 147.51 75.34 0.
171.58 71.71 166.77 48.80 0.
176.19 67.44 168.77 73.00 0.
180.73 77.22 150.06 52.37 0.
159.13 77.98 159.62 57.95 0.
132.10 55.13 166.45 62.90 0.
183.90 100.64 169.87 75.34 0.
157.81 58.50 158.50 47.52 0.
190.32 71.38 163.22 66.25 0.
181.31 58.99 151.34 59.23 0.
157.81 74.71 164.07 87.54 0.
187.11 93.16 145.54 43.16 0.
178.08 52.13 173.39 53.63 0.
161.62 63.21 146.25 55.18 0.
187.11 72.69 155.23 50.46 0.
167.75 97.10 186.59 78.28 0.
163.20 55.76 171.50 52.37 0.
165.12 70.07 169.13 47.52 0.
162.42 82.30 169.87 63.50 0.
182.55 82.80 165.54 47.52 0.
187.11 77.59 155.86 70.25 0.
177.12 51.21 147.51 62.90 0.
177.59 47.64 149.12 69.54 0.
183.20 97.10 149.12 41.89 0.
150.10 78.76 151.34 46.81 0.
189.14 56.93 176.46 76.42 0.
174.87 55.76 140.31 59.86 0.
185.40 58.50 157.35 50.46 0.
170.03 83.33 151.74 70.25 0.
169.65 81.81 152.87 69.89 0.
161.22 68.11 165.84 78.28 0.
168.51 51.21 172.88 62.00 0.
178.08 68.11 157.35 69.19 0.
155.92 56.36 162.94 73.88 0.
159.56 62.82 142.91 82.64 0.
150.10 82.80 170.66 66.57 0.
180.17 55.76 150.50 47.52 0.
176.65 74.03 152.13 49.38 0.
166.25 68.77 157.64 49.38 0.
185.40 63.58 149.12 69.54 0.
154.90 66.43 157.64 64.41 0.
201.90 54.46 147.51 52.37 0.
168.13 77.22 161.28 69.89 0.
182.55 80.88 180.26 61.39 0.
153.27 77.22 159.06 70.25 0.
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172.38 91.78 156.16 55.91 0.
142.33 72.03 181.69 54.81 0.
169.26 54.46 172.40 72.17 0.
180.17 79.58 152.13 57.29 0.
170.03 79.58 164.94 46.04 0.
169.65 76.48 166.14 58.60 0.
158.26 66.09 183.59 49.38 0.
140.74 72.36 172.88 46.81 0.
170.03 72.03 165.24 69.19 0.
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186.22 72.03 143.90 52.37 0.
164.74 80.44 150.06 68.85 0.
195.21 55.13 170.26 64.11 0.
170.03 53.74 149.60 64.41 0.
154.38 62.43 162.66 70.62 0.
170.03 68.44 145.54 64.41 0.
178.58 94.85 164.94 62.90 0.
184.62 76.12 186.59 50.46 0.
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156.41 58.00 160.17 70.62 0.
154.90 47.64 150.50 43.16 0.
174.87 74.71 164.36 62.00 0.
167.38 40.16 157.35 54.81 0.
174.02 90.61 181.69 67.20 0.
189.14 77.22 153.23 73.88 0.
178.08 72.36 170.26 67.20 0.
178.58 61.63 167.41 44.24 0.
162.81 71.38 162.10 34.98 0.
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150.80 89.59 172.40 75.34 0.
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155.42 68.44 153.23 65.02 0.
163.59 47.64 151.74 70.25 0.
171.19 78.76 152.87 54.03 0.
172.38 76.12 162.10 68.18 0.
164.35 76.12 164.94 77.62 0.
191.67 75.06 152.13 48.80 0.
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170.80 83.87 149.60 69.89 0.
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140.74 69.42 145.54 72.58 0.
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156.41 49.02 175.10 67.85 0.
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164.35 68.44 162.94 51.92 0.
151.45 88.67 164.65 49.94 0.
190.32 59.46 169.13 64.11 0.
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138.79 64.68 176.46 50.97 0.
153.27 67.44 159.62 73.00 0.
153.83 91.78 173.92 85.47 0.
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153.83 81.34 144.77 54.81 0.
148.60 73.36 166.45 62.60 0.
155.92 83.33 153.23 49.38 0.
168.88 65.04 161.83 73.00 0.
152.08 80.44 172.88 59.86 0.
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175.30 62.82 156.16 62.30 0.
152.69 83.87 141.74 76.42 0.
161.22 73.36 159.06 54.42 0.
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164.35 64.68 156.46 51.45 0.
153.27 65.39 140.31 52.37 0.
159.98 59.92 148.08 66.25 0.
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169.65 83.33 154.59 51.45 0.
164.74 62.04 155.55 53.22 0.
174.44 66.43 169.87 81.56 0.
163.59 100.64 174.49 80.61 0.
153.27 73.02 151.74 67.85 0.
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177.59 76.12 161.00 60.48 0.
156.88 71.05 156.46 52.37 0.
191.67 69.09 166.45 63.20 0.
159.56 59.46 168.77 62.60 0.
163.20 70.73 163.22 68.51 0.
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148.60 51.21 161.28 70.62 0.
178.08 65.74 151.34 52.37 0.
195.21 73.36 169.49 70.25 0.
197.81 52.13 143.90 67.20 0.
164.74 45.95 150.50 54.81 0.
156.88 50.19 152.87 69.54 0.
158.70 100.64 169.87 46.04 0.
170.41 66.09 141.74 75.86 0.
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144.86 97.10 164.65 61.39 0.
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183.20 80.88 155.55 75.86 0.
168.88 77.98 135.41 52.80 0.
168.13 81.34 153.58 49.38 0.
158.26 81.34 161.55 45.19 0.
150.80 73.36 186.59 61.69 0.
159.13 56.93 163.50 54.03 0.
154.38 87.83 162.66 45.19 0.
164.74 65.04 150.06 67.20 0.
170.03 62.82 138.41 56.95 0.
201.90 56.93 155.86 61.69 0.
161.62 47.64 150.50 40.33 0.
201.90 72.36 140.31 38.26 0.
132.10 66.09 169.13 85.47 0.
195.21 43.70 151.74 64.71 0.
176.19 79.17 144.77 73.43 0.
176.19 85.04 180.26 57.95 0.
178.08 58.00 168.08 72.58 0.
179.10 91.78 172.88 48.18 0.
181.31 81.81 154.91 55.18 0.
178.58 87.06 142.91 68.18 0.
197.81 52.13 159.90 68.85 0.
147.78 56.93 148.08 43.16 0.
179.62 68.11 159.06 64.41 0.
163.97 77.22 159.34 57.29 0.
183.90 61.22 158.78 38.26 0.
167.75 52.13 155.23 46.04 0.
183.90 54.46 150.06 40.33 0.
172.38 72.03 169.87 62.00 0.
157.35 77.22 154.59 70.25 0.
161.62 45.95 176.46 69.19 0.
138.79 87.06 171.94 72.17 0.
140.74 53.74 157.93 57.29 0.
167.38 72.03 163.50 78.99 0.
157.81 73.36 135.41 80.61 0.
160.40 72.36 149.60 72.58 0.
150.80 71.71 151.34 53.63 0.
148.60 64.32 160.72 41.89 0.
161.62 89.59 172.88 52.37 0.
140.74 83.87 170.66 87.54 0.
195.21 85.67 168.08 56.61 0.
175.30 79.58 167.41 90.82 0.
150.10 84.44 157.35 52.80 0.
183.20 74.03 147.51 68.51 0.
183.90 84.44 164.36 53.22 0.
154.38 80.01 169.87 49.38 0.
165.12 61.22 140.31 65.02 0.
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167.75 71.05 162.66 57.62 0.
183.20 83.33 151.74 78.99 0.
168.13 56.36 165.54 52.37 0.
184.62 76.85 179.09 78.28 0.
132.10 58.50 174.49 73.88 0.
167.38 58.50 173.92 38.26 0.
173.19 40.16 162.66 74.83 0.
170.80 77.98 148.08 60.17 0.
188.08 49.02 143.90 45.19 0.
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146.89 56.93 180.26 62.60 0.
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138.79 75.06 167.74 69.54 0.
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201.90 68.77 153.58 55.18 0.
188.08 76.48 157.06 71.77 0.
145.92 58.00 159.90 66.57 0.
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197.81 66.09 146.90 63.50 0.
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154.90 79.58 164.65 73.43 0.
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163.97 70.40 168.08 63.80 0.
151.45 71.38 149.12 64.41 0.
174.87 54.46 154.26 75.86 0.
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165.49 54.46 170.66 65.32 0.
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143.68 67.44 162.10 48.18 0.
197.81 54.46 149.12 46.81 0.
156.88 69.09 151.34 46.04 0.
170.03 76.12 173.92 72.17 0.
186.22 69.75 175.10 75.86 0.
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143.68 85.67 155.86 51.92 0.
167.75 87.83 165.54 76.42 0.
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165.49 75.41 154.91 73.00 0.
153.83 93.16 154.91 63.20 0.
148.60 59.46 178.10 66.57 0.
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175.30 64.68 175.75 63.80 0.
142.33 75.76 171.94 51.45 0.
156.41 87.83 163.22 54.81 0.
201.90 72.03 169.13 59.55 0.
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184.62 87.83 169.87 69.19 0.
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170.03 71.05 162.38 55.18 0.
170.80 79.58 160.72 90.82 0.
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170.80 81.81 159.34 54.81 0.
162.02 59.46 154.26 73.00 0.
157.81 85.04 164.07 77.00 0.
159.98 74.37 175.75 63.50 0.
193.26 69.42 169.13 49.94 0.
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168.88 85.04 156.46 63.50 0.
189.14 51.21 166.45 59.23 0.
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157.81 57.47 172.88 69.89 0.
168.88 60.79 147.51 87.54 0.
143.68 82.30 162.38 72.58 0.
163.97 75.06 180.26 75.86 0.
171.98 49.02 157.35 62.00 0.
148.60 63.21 154.91 71.38 0.
166.62 68.11 167.41 54.03 0.
165.49 70.40 157.93 87.54 0.
167.38 74.71 159.34 52.37 0.
180.73 76.85 157.64 61.09 0.
132.10 73.02 158.50 53.63 0.
193.26 85.04 163.79 63.80 0.
152.69 51.21 162.38 82.64 0.
146.89 51.21 181.69 49.94 0.
189.14 80.44 166.77 50.97 0.
142.33 68.77 151.74 85.47 0.
146.89 65.04 173.39 65.32 0.
201.90 80.88 145.54 70.62 0.
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140.74 66.43 169.49 59.86 0.
159.98 77.98 171.50 63.20 0.
180.17 71.71 151.34 54.03 0.
174.87 71.71 186.59 64.71 0.
190.32 97.10 158.21 46.81 0.
174.44 69.09 135.41 61.39 0.
150.10 80.44 170.66 82.64 0.
161.22 74.71 143.90 72.17 0.
166.25 58.50 149.12 68.85 0.
170.80 56.93 140.31 71.38 0.
170.41 84.44 150.93 46.04 0.
138.79 80.01 177.23 77.62 0.
159.56 59.92 167.74 75.86 0.
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155.92 71.71 165.54 59.55 0.
176.19 64.32 159.34 55.18 0.
168.51 85.04 162.38 50.97 0.
161.22 63.95 157.35 81.56 0.
144.86 74.03 143.90 75.86 0.
190.32 82.30 155.55 60.48 0.
171.19 64.32 178.10 46.04 0.
165.12 67.11 157.64 43.16 0.
171.58 63.58 173.92 66.88 0.
169.65 50.19 154.91 73.88 0.
158.70 43.70 150.50 56.95 0.
154.90 89.59 160.45 67.52 0.
157.35 79.58 181.69 58.60 0.
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170.80 84.44 155.55 53.22 0.
159.98 94.85 142.91 51.92 0.
154.90 80.88 145.54 66.57 0.
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162.81 80.44 145.54 57.29 0.
179.62 47.64 140.31 80.61 0.
171.98 45.95 186.59 65.94 0.
138.79 87.83 153.23 51.45 0.
144.86 52.13 161.55 72.58 0.
153.27 40.16 138.41 55.55 0.
180.17 81.81 162.94 74.83 0.
154.90 69.42 160.72 70.62 0.
189.14 74.71 181.69 73.00 0.
153.83 69.42 166.45 78.28 0.
172.78 70.73 178.10 62.30 0.
183.90 86.34 150.93 45.19 0.
172.38 85.04 176.46 79.76 0.
165.49 81.81 150.06 56.95 0.
175.74 62.82 165.24 45.19 0.
157.35 69.42 166.45 80.61 0.
168.88 55.13 152.87 66.25 0.
167.00 52.97 166.77 47.52 0.
173.19 59.92 158.50 80.61 0.
162.02 71.05 155.55 73.43 0.
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136.19 73.02 166.14 80.61 0.
154.38 73.02 171.50 76.42 0.
183.20 67.78 146.90 68.85 0.
169.65 89.59 173.92 67.52 0.
168.88 68.44 172.88 58.28 0.
171.19 68.44 160.72 45.19 0.
195.21 86.34 158.50 61.69 0.
164.74 93.16 169.49 48.80 0.
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136.19 82.30 163.50 53.63 0.
172.78 83.87 160.17 61.69 0.
156.88 73.69 160.45 67.20 0.
143.68 73.02 172.88 52.37 0.
174.44 72.69 157.06 43.16 0.
153.27 45.95 155.86 54.81 0.
153.83 83.87 152.51 53.22 0.
154.38 100.64 152.13 57.95 0.
174.02 70.07 147.51 64.41 0.
177.59 57.47 152.51 76.42 0.
156.41 75.41 147.51 54.81 0.
193.26 85.67 150.93 64.71 0.
162.81 91.78 172.88 72.17 0.
191.67 52.97 173.39 57.62 0.
177.59 76.12 159.34 46.04 0.
175.30 80.44 181.69 58.92 0.