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Introduction to Monte Carlo Methods

Ajay Rawat

 

# Monte Carlo Integration

import math
import numpy as np
import matplotlib.pyplot as plt
a = 0.                      # Lower limit
b = math.pi                 # Upper limit  
Trials = 10000              # Number of sampling nodes
f = lambda x: 0.5*sin(x)    # function you want to integrate
value = []
for j in range (1, 101):
    print j
    xsum = 0.
    for i in range (1, Trials):
        x = (b-a)*np.random.rand() + a
        y = f(x)
        xsum += y
    ans =  (b-a)*xsum/(Trials - 1)
    value.append(ans)
    print 'Monte carlo Integration value:' 
    print value
plt.hist(value, bins = 10)
plt.show()
WARNING: Output truncated!
1 Monte carlo Integration value: [0.99807081136671838] 2 Monte carlo Integration value: [0.99807081136671838, 0.9974056045640447] 3 Monte carlo Integration value: [0.99807081136671838, 0.9974056045640447, 0.99900704208724211] 4 Monte carlo Integration value: [0.99807081136671838, 0.9974056045640447, 0.99900704208724211, 1.0039382842548841] 5 Monte carlo Integration value: [0.99807081136671838, 0.9974056045640447, 0.99900704208724211, 1.0039382842548841, 0.99272673569312253] 6 Monte carlo Integration value: [0.99807081136671838, 0.9974056045640447, 0.99900704208724211, 1.0039382842548841, 0.99272673569312253, 1.0017997381195367] 7 Monte carlo Integration value: [0.99807081136671838, 0.9974056045640447, 0.99900704208724211, 1.0039382842548841, 0.99272673569312253, 1.0017997381195367, 1.0019601740538657] 8 Monte carlo Integration value: [0.99807081136671838, 0.9974056045640447, 0.99900704208724211, 1.0039382842548841, 0.99272673569312253, 1.0017997381195367, 1.0019601740538657, 1.0059545476494509] 9 Monte carlo Integration value: [0.99807081136671838, 0.9974056045640447, 0.99900704208724211, 1.0039382842548841, 0.99272673569312253, 1.0017997381195367, 1.0019601740538657, 1.0059545476494509, 0.99326885417660238] 10 Monte carlo Integration value: [0.99807081136671838, 0.9974056045640447, 0.99900704208724211, 1.0039382842548841, 0.99272673569312253, 1.0017997381195367, 1.0019601740538657, 1.0059545476494509, 0.99326885417660238, 0.99336089010300355] 11 Monte carlo Integration value: [0.99807081136671838, 0.9974056045640447, 0.99900704208724211, 1.0039382842548841, 0.99272673569312253, 1.0017997381195367, 1.0019601740538657, 1.0059545476494509, 0.99326885417660238, 0.99336089010300355, 0.99894287672945936] 12 Monte carlo Integration value: [0.99807081136671838, 0.9974056045640447, 0.99900704208724211, 1.0039382842548841, 0.99272673569312253, 1.0017997381195367, 1.0019601740538657, 1.0059545476494509, 0.99326885417660238, 0.99336089010300355, 0.99894287672945936, 0.99383728946629868] 13 Monte carlo Integration value: [0.99807081136671838, 0.9974056045640447, 0.99900704208724211, 1.0039382842548841, 0.99272673569312253, 1.0017997381195367, 1.0019601740538657, 1.0059545476494509, 0.99326885417660238, 0.99336089010300355, 0.99894287672945936, 0.99383728946629868, 0.99687104418402972] 14 Monte carlo Integration value: [0.99807081136671838, 0.9974056045640447, 0.99900704208724211, 1.0039382842548841, 0.99272673569312253, 1.0017997381195367, 1.0019601740538657, 1.0059545476494509, 0.99326885417660238, 0.99336089010300355, 0.99894287672945936, 0.99383728946629868, 0.99687104418402972, 1.0016537085072716] 15 Monte carlo Integration value: [0.99807081136671838, 0.9974056045640447, 0.99900704208724211, 1.0039382842548841, 0.99272673569312253, 1.0017997381195367, 1.0019601740538657, 1.0059545476494509, 0.99326885417660238, 0.99336089010300355, 0.99894287672945936, 0.99383728946629868, 0.99687104418402972, 1.0016537085072716, 0.99243548085248501] 16 Monte carlo Integration value: [0.99807081136671838, 0.9974056045640447, 0.99900704208724211, 1.0039382842548841, 0.99272673569312253, 1.0017997381195367, 1.0019601740538657, 1.0059545476494509, 0.99326885417660238, 0.99336089010300355, 0.99894287672945936, 0.99383728946629868, 0.99687104418402972, 1.0016537085072716, 0.99243548085248501, 1.0007487993999387] 17 Monte carlo Integration value: [0.99807081136671838, 0.9974056045640447, 0.99900704208724211, 1.0039382842548841, 0.99272673569312253, 1.0017997381195367, 1.0019601740538657, 1.0059545476494509, 0.99326885417660238, 0.99336089010300355, 0.99894287672945936, 0.99383728946629868, 0.99687104418402972, 1.0016537085072716, 0.99243548085248501, 1.0007487993999387, 1.0063433339856798] 18 Monte carlo Integration value: [0.99807081136671838, 0.9974056045640447, 0.99900704208724211, 1.0039382842548841, 0.99272673569312253, 1.0017997381195367, 1.0019601740538657, 1.0059545476494509, 0.99326885417660238, 0.99336089010300355, 0.99894287672945936, 0.99383728946629868, 0.99687104418402972, 1.0016537085072716, 0.99243548085248501, 1.0007487993999387, 1.0063433339856798, 0.9980576373773471] 19 Monte carlo Integration value: [0.99807081136671838, 0.9974056045640447, 0.99900704208724211, 1.0039382842548841, 0.99272673569312253, 1.0017997381195367, 1.0019601740538657, 1.0059545476494509, 0.99326885417660238, 0.99336089010300355, 0.99894287672945936, 0.99383728946629868, 0.99687104418402972, 1.0016537085072716, 0.99243548085248501, 1.0007487993999387, 1.0063433339856798, 0.9980576373773471, 1.0033251855870511] 20 Monte carlo Integration value: ... 724211, 1.0039382842548841, 0.99272673569312253, 1.0017997381195367, 1.0019601740538657, 1.0059545476494509, 0.99326885417660238, 0.99336089010300355, 0.99894287672945936, 0.99383728946629868, 0.99687104418402972, 1.0016537085072716, 0.99243548085248501, 1.0007487993999387, 1.0063433339856798, 0.9980576373773471, 1.0033251855870511, 1.0088134130593556, 1.0026672954771716, 0.99598834683680115, 1.0035464433884553, 1.0081943159928541, 0.99619472810364795, 1.0017883550620736, 0.99674941174967946, 1.0016629084932638, 0.9956162123281197, 0.99555104902539293, 1.000551203107755, 0.99667140893815898, 1.0033531990871807, 1.0019152365056347, 0.99961892235979899, 1.0004272619333301, 0.9960109223222976, 0.99245102209994862, 0.9960446391732658, 0.99509038209765244, 1.0051959385854305, 1.0010787437021063, 0.995066845772278, 1.0072653482868903, 0.99949289835523714, 0.99012983730095949, 0.99898031897245776, 1.0003037676440847, 0.99803819186722365, 1.0009608944492439, 0.99431250695270734, 0.99606686010579382, 1.0040025344003531, 1.0057155628669945, 0.99924086908838594, 0.99889843981829851, 0.98992548335186836, 1.0004702592804766, 1.0024955890158935, 0.99850765694576682, 1.0009306793325754, 0.99067721248489271, 0.99986293538614368, 0.99340378770870685, 1.0060729363831871, 1.0026516409919741, 0.99602453125319801, 1.0060371389249325, 0.99615973169266669, 0.99849594736384006, 0.99855786002650226, 0.99224568074622121, 1.0108732086009351, 1.0051945476846076, 0.99762533996529801, 0.99771865349769662, 0.99796935877555004, 1.0096331597698476, 1.0005036196254042, 0.99738154979262672, 0.99709153044906162, 0.98967939608771749, 0.99548663721273356, 0.99974386982776753, 1.0015268404797228, 0.99722125924915839, 0.99213428589663599, 1.0057808300768372, 1.0070093211639013, 1.0018281209047071, 0.99811112154154025, 1.0021351112614054, 0.99725023844852212] 94 Monte carlo Integration value: [0.99807081136671838, 0.9974056045640447, 0.99900704208724211, 1.0039382842548841, 0.99272673569312253, 1.0017997381195367, 1.0019601740538657, 1.0059545476494509, 0.99326885417660238, 0.99336089010300355, 0.99894287672945936, 0.99383728946629868, 0.99687104418402972, 1.0016537085072716, 0.99243548085248501, 1.0007487993999387, 1.0063433339856798, 0.9980576373773471, 1.0033251855870511, 1.0088134130593556, 1.0026672954771716, 0.99598834683680115, 1.0035464433884553, 1.0081943159928541, 0.99619472810364795, 1.0017883550620736, 0.99674941174967946, 1.0016629084932638, 0.9956162123281197, 0.99555104902539293, 1.000551203107755, 0.99667140893815898, 1.0033531990871807, 1.0019152365056347, 0.99961892235979899, 1.0004272619333301, 0.9960109223222976, 0.99245102209994862, 0.9960446391732658, 0.99509038209765244, 1.0051959385854305, 1.0010787437021063, 0.995066845772278, 1.0072653482868903, 0.99949289835523714, 0.99012983730095949, 0.99898031897245776, 1.0003037676440847, 0.99803819186722365, 1.0009608944492439, 0.99431250695270734, 0.99606686010579382, 1.0040025344003531, 1.0057155628669945, 0.99924086908838594, 0.99889843981829851, 0.98992548335186836, 1.0004702592804766, 1.0024955890158935, 0.99850765694576682, 1.0009306793325754, 0.99067721248489271, 0.99986293538614368, 0.99340378770870685, 1.0060729363831871, 1.0026516409919741, 0.99602453125319801, 1.0060371389249325, 0.99615973169266669, 0.99849594736384006, 0.99855786002650226, 0.99224568074622121, 1.0108732086009351, 1.0051945476846076, 0.99762533996529801, 0.99771865349769662, 0.99796935877555004, 1.0096331597698476, 1.0005036196254042, 0.99738154979262672, 0.99709153044906162, 0.98967939608771749, 0.99548663721273356, 0.99974386982776753, 1.0015268404797228, 0.99722125924915839, 0.99213428589663599, 1.0057808300768372, 1.0070093211639013, 1.0018281209047071, 0.99811112154154025, 1.0021351112614054, 0.99725023844852212, 1.002124921677557] 95 Monte carlo Integration value: [0.99807081136671838, 0.9974056045640447, 0.99900704208724211, 1.0039382842548841, 0.99272673569312253, 1.0017997381195367, 1.0019601740538657, 1.0059545476494509, 0.99326885417660238, 0.99336089010300355, 0.99894287672945936, 0.99383728946629868, 0.99687104418402972, 1.0016537085072716, 0.99243548085248501, 1.0007487993999387, 1.0063433339856798, 0.9980576373773471, 1.0033251855870511, 1.0088134130593556, 1.0026672954771716, 0.99598834683680115, 1.0035464433884553, 1.0081943159928541, 0.99619472810364795, 1.0017883550620736, 0.99674941174967946, 1.0016629084932638, 0.9956162123281197, 0.99555104902539293, 1.000551203107755, 0.99667140893815898, 1.0033531990871807, 1.0019152365056347, 0.99961892235979899, 1.0004272619333301, 0.9960109223222976, 0.99245102209994862, 0.9960446391732658, 0.99509038209765244, 1.0051959385854305, 1.0010787437021063, 0.995066845772278, 1.0072653482868903, 0.99949289835523714, 0.99012983730095949, 0.99898031897245776, 1.0003037676440847, 0.99803819186722365, 1.0009608944492439, 0.99431250695270734, 0.99606686010579382, 1.0040025344003531, 1.0057155628669945, 0.99924086908838594, 0.99889843981829851, 0.98992548335186836, 1.0004702592804766, 1.0024955890158935, 0.99850765694576682, 1.0009306793325754, 0.99067721248489271, 0.99986293538614368, 0.99340378770870685, 1.0060729363831871, 1.0026516409919741, 0.99602453125319801, 1.0060371389249325, 0.99615973169266669, 0.99849594736384006, 0.99855786002650226, 0.99224568074622121, 1.0108732086009351, 1.0051945476846076, 0.99762533996529801, 0.99771865349769662, 0.99796935877555004, 1.0096331597698476, 1.0005036196254042, 0.99738154979262672, 0.99709153044906162, 0.98967939608771749, 0.99548663721273356, 0.99974386982776753, 1.0015268404797228, 0.99722125924915839, 0.99213428589663599, 1.0057808300768372, 1.0070093211639013, 1.0018281209047071, 0.99811112154154025, 1.0021351112614054, 0.99725023844852212, 1.002124921677557, 1.0033875594404802] 96 Monte carlo Integration value: [0.99807081136671838, 0.9974056045640447, 0.99900704208724211, 1.0039382842548841, 0.99272673569312253, 1.0017997381195367, 1.0019601740538657, 1.0059545476494509, 0.99326885417660238, 0.99336089010300355, 0.99894287672945936, 0.99383728946629868, 0.99687104418402972, 1.0016537085072716, 0.99243548085248501, 1.0007487993999387, 1.0063433339856798, 0.9980576373773471, 1.0033251855870511, 1.0088134130593556, 1.0026672954771716, 0.99598834683680115, 1.0035464433884553, 1.0081943159928541, 0.99619472810364795, 1.0017883550620736, 0.99674941174967946, 1.0016629084932638, 0.9956162123281197, 0.99555104902539293, 1.000551203107755, 0.99667140893815898, 1.0033531990871807, 1.0019152365056347, 0.99961892235979899, 1.0004272619333301, 0.9960109223222976, 0.99245102209994862, 0.9960446391732658, 0.99509038209765244, 1.0051959385854305, 1.0010787437021063, 0.995066845772278, 1.0072653482868903, 0.99949289835523714, 0.99012983730095949, 0.99898031897245776, 1.0003037676440847, 0.99803819186722365, 1.0009608944492439, 0.99431250695270734, 0.99606686010579382, 1.0040025344003531, 1.0057155628669945, 0.99924086908838594, 0.99889843981829851, 0.98992548335186836, 1.0004702592804766, 1.0024955890158935, 0.99850765694576682, 1.0009306793325754, 0.99067721248489271, 0.99986293538614368, 0.99340378770870685, 1.0060729363831871, 1.0026516409919741, 0.99602453125319801, 1.0060371389249325, 0.99615973169266669, 0.99849594736384006, 0.99855786002650226, 0.99224568074622121, 1.0108732086009351, 1.0051945476846076, 0.99762533996529801, 0.99771865349769662, 0.99796935877555004, 1.0096331597698476, 1.0005036196254042, 0.99738154979262672, 0.99709153044906162, 0.98967939608771749, 0.99548663721273356, 0.99974386982776753, 1.0015268404797228, 0.99722125924915839, 0.99213428589663599, 1.0057808300768372, 1.0070093211639013, 1.0018281209047071, 0.99811112154154025, 1.0021351112614054, 0.99725023844852212, 1.002124921677557, 1.0033875594404802, 0.99789772478699401] 97 Monte carlo Integration value: [0.99807081136671838, 0.9974056045640447, 0.99900704208724211, 1.0039382842548841, 0.99272673569312253, 1.0017997381195367, 1.0019601740538657, 1.0059545476494509, 0.99326885417660238, 0.99336089010300355, 0.99894287672945936, 0.99383728946629868, 0.99687104418402972, 1.0016537085072716, 0.99243548085248501, 1.0007487993999387, 1.0063433339856798, 0.9980576373773471, 1.0033251855870511, 1.0088134130593556, 1.0026672954771716, 0.99598834683680115, 1.0035464433884553, 1.0081943159928541, 0.99619472810364795, 1.0017883550620736, 0.99674941174967946, 1.0016629084932638, 0.9956162123281197, 0.99555104902539293, 1.000551203107755, 0.99667140893815898, 1.0033531990871807, 1.0019152365056347, 0.99961892235979899, 1.0004272619333301, 0.9960109223222976, 0.99245102209994862, 0.9960446391732658, 0.99509038209765244, 1.0051959385854305, 1.0010787437021063, 0.995066845772278, 1.0072653482868903, 0.99949289835523714, 0.99012983730095949, 0.99898031897245776, 1.0003037676440847, 0.99803819186722365, 1.0009608944492439, 0.99431250695270734, 0.99606686010579382, 1.0040025344003531, 1.0057155628669945, 0.99924086908838594, 0.99889843981829851, 0.98992548335186836, 1.0004702592804766, 1.0024955890158935, 0.99850765694576682, 1.0009306793325754, 0.99067721248489271, 0.99986293538614368, 0.99340378770870685, 1.0060729363831871, 1.0026516409919741, 0.99602453125319801, 1.0060371389249325, 0.99615973169266669, 0.99849594736384006, 0.99855786002650226, 0.99224568074622121, 1.0108732086009351, 1.0051945476846076, 0.99762533996529801, 0.99771865349769662, 0.99796935877555004, 1.0096331597698476, 1.0005036196254042, 0.99738154979262672, 0.99709153044906162, 0.98967939608771749, 0.99548663721273356, 0.99974386982776753, 1.0015268404797228, 0.99722125924915839, 0.99213428589663599, 1.0057808300768372, 1.0070093211639013, 1.0018281209047071, 0.99811112154154025, 1.0021351112614054, 0.99725023844852212, 1.002124921677557, 1.0033875594404802, 0.99789772478699401, 0.99032283374883601] 98 Monte carlo Integration value: [0.99807081136671838, 0.9974056045640447, 0.99900704208724211, 1.0039382842548841, 0.99272673569312253, 1.0017997381195367, 1.0019601740538657, 1.0059545476494509, 0.99326885417660238, 0.99336089010300355, 0.99894287672945936, 0.99383728946629868, 0.99687104418402972, 1.0016537085072716, 0.99243548085248501, 1.0007487993999387, 1.0063433339856798, 0.9980576373773471, 1.0033251855870511, 1.0088134130593556, 1.0026672954771716, 0.99598834683680115, 1.0035464433884553, 1.0081943159928541, 0.99619472810364795, 1.0017883550620736, 0.99674941174967946, 1.0016629084932638, 0.9956162123281197, 0.99555104902539293, 1.000551203107755, 0.99667140893815898, 1.0033531990871807, 1.0019152365056347, 0.99961892235979899, 1.0004272619333301, 0.9960109223222976, 0.99245102209994862, 0.9960446391732658, 0.99509038209765244, 1.0051959385854305, 1.0010787437021063, 0.995066845772278, 1.0072653482868903, 0.99949289835523714, 0.99012983730095949, 0.99898031897245776, 1.0003037676440847, 0.99803819186722365, 1.0009608944492439, 0.99431250695270734, 0.99606686010579382, 1.0040025344003531, 1.0057155628669945, 0.99924086908838594, 0.99889843981829851, 0.98992548335186836, 1.0004702592804766, 1.0024955890158935, 0.99850765694576682, 1.0009306793325754, 0.99067721248489271, 0.99986293538614368, 0.99340378770870685, 1.0060729363831871, 1.0026516409919741, 0.99602453125319801, 1.0060371389249325, 0.99615973169266669, 0.99849594736384006, 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