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HSG-MCS-HS21_Julia/Problemsets/Solutions/PS03_CLT_Solution.ipynb

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"data": {
"text/plain": [
"printyellow (generic function with 1 method)"
]
},
"execution_count": 1,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"using Printf, Statistics, Random, Distributions\n",
"include(\"jlFiles/printmat.jl\")"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [],
"source": [
"using Plots\n",
"\n",
"gr(size=(480,320))\n",
"default(fmt = :svg)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Task 1 (done)\n",
"\n",
"Generate 10000 values of $x=z-2$ where $z$ has a `Chisq(2)` distribution. Show a histogram (with bins from at least -3 to 6) and comment on whether the distribution looks normal. Add the pdf of $x$. \n",
"\n",
"Notice that the (population) mean and standard deviation of a `Chisq(n)` variable are $\\mu=n$ and $\\sigma=\\sqrt{2n}$."
]
},
{
"cell_type": "code",
"execution_count": 3,
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"estimated mean and std: 0.018 2.027\n",
"\n",
"\u001b[31m\u001b[1mstrongly non-normal\u001b[22m\u001b[39m\n"
]
}
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"source": [
"Random.seed!(123) #so the same random numbers are drawn \n",
" #each time you run this cell\n",
"x = rand(Chisq(2),10_000) .- 2\n",
"\n",
"printlnPs(\"estimated mean and std: \",mean(x),\" \",std(x))\n",
"\n",
"zGrid = range(0,8,length=101)\n",
"\n",
"p1 = histogram( x,bins = -3:0.1:6,\n",
" normalized = true, \n",
" legend=false,\n",
" title = \"Histogram of x\",\n",
" xlabel = \"x\" )\n",
"plot!(zGrid.-2,pdf(Chisq(2),zGrid),linewidth=2) #pdf(x) value equals pdf(z) value, where x=z-2\n",
"display(p1)\n",
"\n",
"printred(\"\\nstrongly non-normal\")"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Task 2: Law of Large Numbers\n",
"\n",
"\n",
"Use the $x$ series above and calculate the average of the first $s$ observations. Do this for $s=1$, $s=2$, ..., $s=2000$. Plot the averages (on the vertical axis) against $s$ (on the horizontal axis).\n",
"\n",
"Hint: `cumsum()` or a loop over `s`."
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [
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" 562.554,299.187 563.349,297.278 564.143,293.279 564.938,291.837 565.732,292.692 566.527,293.868 567.322,293.772 568.116,288.247 568.911,288.909 569.705,289.988 \n",
" 570.5,291.879 571.294,292.749 572.089,289.454 572.883,288.584 573.678,289.426 574.473,291.006 575.267,293.087 576.062,293.033 576.856,290.052 577.651,290.728 \n",
" 578.445,288.694 579.24,288.937 580.035,284.184 580.829,284.15 581.624,284.85 582.418,282.86 583.213,283.563 584.007,282.649 584.802,284.126 585.596,279.416 \n",
" 586.391,278.706 587.186,279.39 587.98,280.905 588.775,279.539 589.569,277.691 590.364,279.648 591.158,277.481 591.953,273.422 592.747,272.748 593.542,274.299 \n",
" 594.337,274.17 595.131,275.361 595.926,277.228 596.72,275.948 597.515,268.265 598.309,269.371 599.104,270.913 599.898,267.365 600.693,268.791 601.488,268.736 \n",
" 602.282,270.643 603.077,272.491 603.871,269.397 604.666,267.53 605.46,268.04 606.255,267.506 607.049,265.341 607.844,266.993 608.639,265.511 609.433,264.277 \n",
" 610.228,264.422 611.022,264.659 611.817,265.743 612.611,264.983 613.406,264.361 614.2,262.518 614.995,262.091 615.79,263.392 616.584,258.109 617.379,257.97 \n",
" 618.173,259.442 618.968,256.278 619.762,247.536 620.557,242.715 621.352,244.508 622.146,245.658 622.941,247.197 623.735,248.26 624.53,242.163 625.324,241.772 \n",
" 626.119,241.138 626.913,238.823 627.708,240.585 628.503,240.891 629.297,242.032 630.092,236.166 630.886,236.067 631.681,236.393 632.475,237.526 633.27,238.711 \n",
" 634.064,237.316 634.859,235.198 635.654,236.588 636.448,238.41 637.243,240.049 638.037,237.917 638.832,238.058 639.626,239.262 640.421,239.441 641.215,240.894 \n",
" 642.01,242.009 642.805,243.775 643.599,243.24 644.394,243.626 645.188,242.096 645.983,243.605 646.777,244.425 647.572,241.62 648.366,239.219 649.161,237.021 \n",
" 649.956,238.76 650.75,237.402 651.545,238.742 652.339,236.09 653.134,237.839 653.928,237.021 654.723,237.161 655.517,238.83 656.312,240.471 657.107,241.152 \n",
" 657.901,242.588 658.696,243.05 659.49,244.47 660.285,243.936 661.079,245.626 661.874,246.528 662.668,246.884 663.463,247.209 664.258,247.948 665.052,245.037 \n",
" 665.847,245.07 666.641,241.38 667.436,242.428 668.23,243.402 669.025,243.416 669.82,242.499 670.614,240.321 671.409,241.012 672.203,242.561 672.998,243.11 \n",
" 673.792,243.131 674.587,244.069 675.381,245.132 676.176,241.412 676.971,240.201 677.765,240.45 678.56,241.734 679.354,242.609 680.149,243.841 680.943,243.597 \n",
" 681.738,244.177 682.532,245.695 683.327,242.287 684.122,243.344 684.916,244.953 685.711,244.505 686.505,244.285 687.3,244.356 688.094,244.83 688.889,245.551 \n",
" 689.683,246.161 690.478,241.071 691.273,242.278 692.067,243.822 692.862,245.427 693.656,246.956 694.451,248.358 695.245,246.286 696.04,246.349 696.834,246.172 \n",
" 697.629,247.413 698.424,245.713 699.218,245.603 700.013,246.357 700.807,247.655 701.602,247.672 702.396,248.23 703.191,249.678 703.985,248.588 704.78,248.137 \n",
" 705.575,248.844 706.369,250.39 707.164,251.079 707.958,250.259 708.753,249.343 709.547,248.749 710.342,248.777 711.137,248.82 711.931,248.897 712.726,247.605 \n",
" 713.52,247.155 714.315,247.205 715.109,248.668 715.904,250.041 716.698,251.221 717.493,251.989 718.288,253.361 719.082,252.999 719.877,251.211 720.671,251.889 \n",
" 721.466,253.266 722.26,253.061 723.055,252.966 723.849,252.125 724.644,253.625 725.439,253.622 726.233,252.167 727.028,252.184 727.822,249.89 728.617,250.656 \n",
" 729.411,247.772 730.206,249.207 731,249.707 731.795,250.421 732.59,250.691 733.384,251.04 734.179,252.344 734.973,252.757 735.768,252.534 736.562,253.154 \n",
" 737.357,253.709 738.151,251.059 738.946,250.997 739.741,252.459 740.535,252.245 741.33,251.912 742.124,252.946 742.919,253.918 743.713,252.152 744.508,250.072 \n",
" 745.302,249.347 746.097,249.041 746.892,249.997 747.686,251.366 748.481,250.65 749.275,249.139 750.07,249.348 750.864,250.167 751.659,251.41 752.454,250.241 \n",
" 753.248,249.181 754.043,247.19 754.837,247.373 755.632,248.381 756.426,243.244 757.221,244.265 758.015,245.521 758.81,246.611 759.605,247.67 760.399,247.417 \n",
" 761.194,248.013 761.988,247.761 762.783,242.168 763.577,242.582 764.372,243.804 765.166,244.98 765.961,245.783 766.756,246.903 767.55,247.752 768.345,248.919 \n",
" 769.139,248.816 769.934,247.922 770.728,245.069 771.523,246.358 772.317,247.526 773.112,248.816 773.907,249.097 774.701,250.117 775.496,250.843 776.29,251.282 \n",
" 777.085,250.43 777.879,246.452 778.674,247.341 779.468,246.515 780.263,247.246 781.058,246.818 781.852,246.455 782.647,246.072 783.441,245.377 784.236,246.144 \n",
" 785.03,247.178 785.825,244.358 786.619,245.526 787.414,246.717 788.209,247.035 789.003,245.123 789.798,246.044 790.592,246.969 791.387,247.241 792.181,243.991 \n",
" 792.976,244.551 793.771,244.701 794.565,245.588 795.36,246.384 796.154,244.767 796.949,245.579 797.743,246.851 798.538,248.065 799.332,246.984 800.127,248.187 \n",
" 800.922,249.351 801.716,250.538 802.511,250.096 803.305,251.068 804.1,251.501 804.894,252.073 805.689,252.768 806.483,253.608 807.278,251.871 808.073,252.603 \n",
" 808.867,251.963 809.662,252.785 810.456,253.766 811.251,254.105 812.045,253.095 812.84,253.573 813.634,253.718 814.429,254.845 815.224,256.07 816.018,257.16 \n",
" 816.813,257.873 817.607,258.277 818.402,259.459 819.196,260.722 819.991,261.889 820.785,262.244 821.58,262.59 822.375,263.596 823.169,264.759 823.964,264.003 \n",
" 824.758,264.397 825.553,265.374 826.347,263.888 827.142,264.429 827.936,264.848 828.731,265.814 829.526,263.995 830.32,264.756 831.115,261.915 831.909,262.826 \n",
" 832.704,263.19 833.498,261.245 834.293,261.994 835.087,261.196 835.882,260.426 836.677,256.33 837.471,257.548 838.266,256.294 839.06,257.503 839.855,258.714 \n",
" 840.649,259.799 841.444,260.092 842.239,260.056 843.033,260.586 843.828,261.296 844.622,260.701 845.417,261.238 846.211,261.528 847.006,262.16 847.8,259.741 \n",
" 848.595,258.546 849.39,259.37 850.184,260.487 850.979,260.839 851.773,260.437 852.568,256.66 853.362,257.466 854.157,257.348 854.951,258.545 855.746,255.246 \n",
" 856.541,254.016 857.335,255.026 858.13,255.529 858.924,256.092 859.719,256.405 860.513,257.532 861.308,258.354 862.102,259.441 862.897,259.509 863.692,260.502 \n",
" 864.486,256.414 865.281,257.472 866.075,258.225 866.87,258.377 867.664,259.185 868.459,260.035 869.253,260.835 870.048,261.885 870.843,261.294 871.637,261.766 \n",
" 872.432,258.015 873.226,258.707 874.021,259.557 874.815,260.163 875.61,261.016 876.404,261.606 877.199,262.539 877.994,262.322 878.788,261.743 879.583,262.658 \n",
" 880.377,263.349 881.172,262.32 881.966,262.961 882.761,263.256 883.556,262.399 884.35,262.702 885.145,263.527 885.939,263.757 886.734,264.03 887.528,264.666 \n",
" 888.323,265.127 889.117,265.825 889.912,264.797 890.707,265.223 891.501,265.06 892.296,265.32 893.09,266.357 893.885,266.077 894.679,265.787 895.474,262.185 \n",
" 896.268,263.078 897.063,263.152 897.858,263.627 898.652,256.4 899.447,254.942 900.241,255.338 901.036,255.927 901.83,256.089 902.625,256.528 903.419,256.583 \n",
" 904.214,256.741 905.009,257.187 905.803,258.197 906.598,258.83 907.392,259.717 908.187,259.118 908.981,259.476 909.776,260.034 910.57,260.761 911.365,259.794 \n",
" 912.16,258.876 912.954,259.525 913.749,260.128 914.543,260.537 915.338,259.914 916.132,259.88 916.927,260.501 917.721,260.8 918.516,260.45 919.311,259.19 \n",
" 920.105,260.016 920.9,259.824 921.694,260.111 922.489,260.248 923.283,260.805 924.078,260.359 924.873,260.133 925.667,261.092 926.462,260.596 927.256,258.83 \n",
" 928.051,259.649 928.845,260.571 929.64,259.756 930.434,258.876 931.229,259.141 932.024,260.104 932.818,260.639 933.613,260.595 934.407,259.223 935.202,258.478 \n",
" 935.996,258.193 936.791,258.213 937.585,256.75 938.38,257.603 939.175,258.371 939.969,257.385 940.764,256.306 941.558,255.284 942.353,256.208 943.147,255.916 \n",
" 943.942,255.184 944.736,255.92 945.531,255.06 946.326,255.697 947.12,256.6 947.915,257.51 948.709,257.427 949.504,257.363 950.298,258.376 951.093,257.213 \n",
" 951.887,256.989 952.682,257.722 953.477,256.925 954.271,256.137 955.066,256.888 955.86,257.318 956.655,257.814 957.449,258.303 958.244,258.524 959.038,258.686 \n",
" 959.833,258.955 960.628,257.121 961.422,256.853 962.217,256.438 963.011,255.685 963.806,255.422 964.6,255.93 965.395,255.873 966.19,256.334 966.984,257.309 \n",
" 967.779,258.16 968.573,258.291 969.368,258.856 970.162,259.843 970.957,260.626 971.751,260.268 972.546,260.936 973.341,261.757 974.135,262.262 974.93,263.089 \n",
" 975.724,262.951 976.519,262.722 977.313,262.822 978.108,263.469 978.902,263.173 979.697,263.089 980.492,263.392 981.286,263.045 982.081,261.931 982.875,261.275 \n",
" 983.67,261.332 984.464,262.131 985.259,261.346 986.053,259.321 986.848,259.673 987.643,259.908 988.437,259.11 989.232,258.291 990.026,258.805 990.821,259.654 \n",
" 991.615,258.788 992.41,259.025 993.204,259.652 993.999,260.272 994.794,259.611 995.588,260.305 996.383,261.229 997.177,261.813 997.972,262.573 998.766,262.33 \n",
" 999.561,262.986 1000.36,263.764 1001.15,263.145 1001.94,263.697 1002.74,264.34 1003.53,264.649 1004.33,265.297 1005.12,265.504 1005.92,265.742 1006.71,266.591 \n",
" 1007.51,267.541 1008.3,267.664 1009.1,266.407 1009.89,266.306 1010.68,266.457 1011.48,265.236 1012.27,265.647 1013.07,265.113 1013.86,266.041 1014.66,266.839 \n",
" 1015.45,267.358 1016.25,267.569 1017.04,267.986 1017.84,268.748 1018.63,266.395 1019.42,267.139 1020.22,264.972 1021.01,265.903 1021.81,265.592 1022.6,265.502 \n",
" 1023.4,265.751 1024.19,266.086 1024.99,266.484 1025.78,265.637 1026.58,265.48 1027.37,265.376 1028.16,265.539 1028.96,264.558 1029.75,264.983 1030.55,265.815 \n",
" 1031.34,265.937 1032.14,264.307 1032.93,264.666 1033.73,265.224 1034.52,262.27 1035.32,262.655 1036.11,262.845 1036.91,262.524 1037.7,262.209 1038.49,262.981 \n",
" 1039.29,262.244 1040.08,260.767 1040.88,260.64 1041.67,261.509 1042.47,260.169 1043.26,260.182 1044.06,260.582 1044.85,260.881 1045.65,261.7 1046.44,261.991 \n",
" 1047.23,260.22 1048.03,260.939 1048.82,261.44 1049.62,262.299 1050.41,262.878 1051.21,263.399 1052,264.247 1052.8,264.922 1053.59,265.75 1054.39,266.335 \n",
" 1055.18,264.969 1055.97,265.142 1056.77,265.167 1057.56,265.899 1058.36,266.366 1059.15,264.984 1059.95,265.392 1060.74,265.918 1061.54,265.77 1062.33,266.323 \n",
" 1063.13,266.558 1063.92,264.277 1064.71,265.135 1065.51,265.335 1066.3,265.099 1067.1,265.943 1067.89,266.569 1068.69,266.396 1069.48,266.615 1070.28,266.675 \n",
" 1071.07,267.182 1071.87,267.141 1072.66,267.769 1073.45,268.177 1074.25,268.105 1075.04,268.647 1075.84,269.317 1076.63,270.033 1077.43,266.689 1078.22,264.643 \n",
" 1079.02,264.981 1079.81,265.342 1080.61,265.664 1081.4,265.869 1082.19,266.57 1082.99,267.27 1083.78,268.11 1084.58,268.841 1085.37,269.346 1086.17,269.411 \n",
" 1086.96,269.341 1087.76,269.525 1088.55,269.998 1089.35,270.827 1090.14,271.659 1090.94,272.089 1091.73,270.568 1092.52,271.336 1093.32,270.812 1094.11,271.053 \n",
" 1094.91,268.845 1095.7,269.389 1096.5,270.203 1097.29,269.595 1098.09,268.817 1098.88,267.693 1099.68,268.5 1100.47,268.958 1101.26,269.329 1102.06,269.861 \n",
" 1102.85,270.032 1103.65,269.654 1104.44,270.285 1105.24,269.354 1106.03,270.173 1106.83,270.804 1107.62,270.446 1108.42,270.587 1109.21,270.958 1110,271.77 \n",
" 1110.8,271.858 1111.59,272.572 1112.39,272.447 1113.18,273.077 1113.98,273.669 1114.77,274.15 1115.57,274.435 1116.36,275.104 1117.16,275.206 1117.95,275.247 \n",
" 1118.74,273.95 1119.54,274.216 1120.33,273.404 1121.13,273.724 1121.92,273.491 1122.72,274.266 1123.51,274.609 1124.31,275.139 1125.1,275.882 1125.9,274.648 \n",
" 1126.69,275.4 1127.48,275.525 1128.28,275.6 1129.07,275.25 1129.87,276.013 1130.66,276.55 1131.46,277.096 1132.25,276.838 1133.05,277.089 1133.84,275.648 \n",
" 1134.64,275.798 1135.43,276.254 1136.22,276.706 1137.02,276.616 1137.81,273.769 1138.61,274.298 1139.4,274.561 1140.2,274.472 1140.99,274.823 1141.79,275.55 \n",
" 1142.58,275.425 1143.38,274.351 1144.17,274.855 1144.96,274.676 1145.76,274.99 1146.55,275.042 1147.35,275.475 1148.14,275.744 1148.94,273.161 1149.73,273.365 \n",
" 1150.53,273.997 1151.32,272.913 1152.12,273.169 1152.91,273.365 1153.71,273.806 1154.5,272.967 1155.29,272.557 1156.09,272.469 1156.88,273.098 1157.68,273.816 \n",
" 1158.47,273.834 1159.27,274.585 1160.06,273.872 1160.86,274.606 1161.65,275.101 1162.45,275.839 1163.24,276.366 1164.03,277.026 1164.83,277.197 1165.62,277.961 \n",
" 1166.42,278.222 1167.21,278.762 1168.01,277.428 1168.8,278.018 1169.6,277.505 1170.39,276.933 1171.19,276.785 1171.98,277.335 1172.77,277.381 1173.57,277.568 \n",
" 1174.36,278.314 1175.16,276.94 1175.95,277.442 1176.75,277.257 1177.54,277.906 1178.34,278.303 1179.13,278.622 1179.93,278.011 1180.72,278.224 1181.51,277.52 \n",
" 1182.31,277.903 1183.1,278.663 1183.9,278.22 1184.69,278.077 1185.49,277.956 1186.28,278.251 1187.08,278.763 1187.87,278.542 1188.67,279.274 1189.46,279.751 \n",
" 1190.25,278.445 1191.05,277.791 1191.84,276.925 1192.64,277.287 1193.43,277.719 1194.23,278.195 1195.02,278.838 1195.82,279.531 1196.61,278.734 1197.41,278.975 \n",
" 1198.2,277.216 1198.99,276.374 1199.79,277.095 1200.58,277.436 1201.38,278.033 1202.17,278.341 1202.97,278.097 1203.76,278.648 1204.56,279.1 1205.35,279.79 \n",
" 1206.15,279.684 1206.94,278.807 1207.74,279.546 1208.53,277.761 1209.32,278.301 1210.12,278.107 1210.91,278.322 1211.71,279.052 1212.5,278.803 1213.3,279.111 \n",
" 1214.09,279.446 1214.89,280.051 1215.68,280.755 1216.48,281.001 1217.27,281.571 1218.06,281.804 1218.86,281.698 1219.65,281.703 1220.45,282.353 1221.24,282.878 \n",
" 1222.04,283.476 1222.83,283.674 1223.63,283.936 1224.42,284.635 1225.22,285.334 1226.01,285.341 1226.8,285.232 1227.6,285.031 1228.39,285.754 1229.19,286.278 \n",
" 1229.98,286.692 1230.78,286.643 1231.57,287.015 1232.37,286.31 1233.16,286.884 1233.96,286.857 1234.75,286.474 1235.54,286.872 1236.34,287.307 1237.13,287.633 \n",
" 1237.93,288.212 1238.72,288.085 1239.52,288.316 1240.31,287.883 1241.11,287.087 1241.9,287.678 1242.7,288.004 1243.49,287.198 1244.28,287.275 1245.08,287.593 \n",
" 1245.87,287.703 1246.67,288.146 1247.46,287.825 1248.26,288.468 1249.05,287.212 1249.85,287.371 1250.64,287.911 1251.44,288.461 1252.23,288.055 1253.02,286.449 \n",
" 1253.82,286.213 1254.61,285.785 1255.41,285.922 1256.2,286.203 1257,286.516 1257.79,286.93 1258.59,286.82 1259.38,287.482 1260.18,287.961 1260.97,288.008 \n",
" 1261.76,287.496 1262.56,286.374 1263.35,286.756 1264.15,287.194 1264.94,287.645 1265.74,287.549 1266.53,287.056 1267.33,287.637 1268.12,288.033 1268.92,286.912 \n",
" 1269.71,287.605 1270.51,287.888 1271.3,288.226 1272.09,288.062 1272.89,288.75 1273.68,288.402 1274.48,288.267 1275.27,288.596 1276.07,288.273 1276.86,288.502 \n",
" 1277.66,288.662 1278.45,289.317 1279.25,289.73 1280.04,289.932 1280.83,290.131 1281.63,290.536 1282.42,290.784 1283.22,288.247 1284.01,288.132 1284.81,288.199 \n",
" 1285.6,287.197 1286.4,287.215 1287.19,287.682 1287.99,288.251 1288.78,288.12 1289.57,288.537 1290.37,288.225 1291.16,288.623 1291.96,288.931 1292.75,286.682 \n",
" 1293.55,287.324 1294.34,287.281 1295.14,286.627 1295.93,286.094 1296.73,284.867 1297.52,283.257 1298.31,283.043 1299.11,283.694 1299.9,282.842 1300.7,283.293 \n",
" 1301.49,282.746 1302.29,283.353 1303.08,282.961 1303.88,283.552 1304.67,283.74 1305.47,282.665 1306.26,283.215 1307.05,283.443 1307.85,282.931 1308.64,282.725 \n",
" 1309.44,280.579 1310.23,281.048 1311.03,278.748 1311.82,278.723 1312.62,278.797 1313.41,278.98 1314.21,279.594 1315,280.019 1315.79,279.321 1316.59,279.273 \n",
" 1317.38,277.103 1318.18,277.532 1318.97,277.492 1319.77,277.684 1320.56,277.578 1321.36,277.897 1322.15,278.489 1322.95,278.227 1323.74,278.692 1324.53,278.748 \n",
" 1325.33,278.168 1326.12,278.514 1326.92,279.027 1327.71,279.314 1328.51,279.655 1329.3,279.015 1330.1,279.03 1330.89,278.166 1331.69,278.643 1332.48,279.18 \n",
" 1333.28,279.314 1334.07,279.396 1334.86,279.444 1335.66,278.865 1336.45,279.184 1337.25,279.157 1338.04,276.274 1338.84,276.77 1339.63,277.385 1340.43,277.76 \n",
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" 1349.17,279.465 1349.96,279.872 1350.76,280.247 1351.55,280.773 1352.34,280.937 1353.14,281.441 1353.93,281.392 1354.73,281.107 1355.52,280.598 1356.32,281.17 \n",
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" 1452.46,285.346 1453.25,285.868 1454.05,286.252 1454.84,286.654 1455.64,286.862 1456.43,286.985 1457.23,286.586 1458.02,287.023 1458.82,287.221 1459.61,287.754 \n",
" 1460.4,287.676 1461.2,287.797 1461.99,287.875 1462.79,288.311 1463.58,288.291 1464.38,288.708 1465.17,288.882 1465.97,288.391 1466.76,288.116 1467.56,288.391 \n",
" 1468.35,288.678 1469.14,289.154 1469.94,289.359 1470.73,289.768 1471.53,289.282 1472.32,288.153 1473.12,287.256 1473.91,287.12 1474.71,286.655 1475.5,286.803 \n",
" 1476.3,286.637 1477.09,286.896 1477.88,286.321 1478.68,285.477 1479.47,285.961 1480.27,286.391 1481.06,286.345 1481.86,286.819 1482.65,287.079 1483.45,284.745 \n",
" 1484.24,284.602 1485.04,284.824 1485.83,285.097 1486.62,285.363 1487.42,282.226 1488.21,282.397 1489.01,282.457 1489.8,281.531 1490.6,281.949 1491.39,281.709 \n",
" 1492.19,280.513 1492.98,281.002 1493.78,281.532 1494.57,281.998 1495.36,281.743 1496.16,282.24 1496.95,282.766 1497.75,283.201 1498.54,283.698 1499.34,284.201 \n",
" 1500.13,284.349 1500.93,284.56 1501.72,284.877 1502.52,285.14 1503.31,285.469 1504.11,285.43 1504.9,285.863 1505.69,285.255 1506.49,285.703 1507.28,284.711 \n",
" 1508.08,285.223 1508.87,285.195 1509.67,285.226 1510.46,285.762 1511.26,285.81 1512.05,286.087 1512.85,283.16 1513.64,283.151 1514.43,282.697 1515.23,282.332 \n",
" 1516.02,281.65 1516.82,282.025 1517.61,281.505 1518.41,279.951 1519.2,279.934 1520,279.072 1520.79,279.059 1521.59,279.334 1522.38,279.824 1523.17,279.195 \n",
" 1523.97,279.659 1524.76,279.687 1525.56,280.152 1526.35,280.689 1527.15,280.848 1527.94,280.929 1528.74,280.212 1529.53,279.485 1530.33,279.068 1531.12,279.117 \n",
" 1531.91,279.299 1532.71,279.457 1533.5,279.751 1534.3,280.132 1535.09,280.687 1535.89,280.272 1536.68,279.863 1537.48,280.394 1538.27,278.604 1539.07,278.517 \n",
" 1539.86,278.656 1540.65,279.139 1541.45,279.443 1542.24,279.772 1543.04,280.234 1543.83,280.506 1544.63,280.959 1545.42,281.318 1546.22,281.577 1547.01,281.963 \n",
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" 1603.42,277.047 1604.22,277.008 1605.01,277.517 1605.81,277.449 1606.6,277.4 1607.4,276.869 1608.19,276.98 1608.99,276.408 1609.78,276.925 1610.58,276.914 \n",
" 1611.37,276.307 1612.16,276.692 1612.96,277.019 1613.75,276.928 1614.55,277.055 1615.34,277.472 1616.14,276.863 1616.93,276.905 1617.73,277.2 1618.52,276.904 \n",
" 1619.32,276.924 1620.11,277.323 1620.9,277.761 1621.7,277.746 1622.49,278.238 1623.29,278.37 1624.08,278.729 1624.88,277.782 1625.67,277.787 1626.47,278.079 \n",
" 1627.26,277.868 1628.06,278.387 1628.85,278.743 1629.65,279.202 1630.44,279.566 1631.23,279.654 1632.03,279.178 1632.82,279.142 1633.62,279.288 1634.41,279.517 \n",
" 1635.21,279.644 1636,280.024 1636.8,280.276 1637.59,280.622 1638.39,280.02 1639.18,279.056 1639.97,279.552 1640.77,279.801 1641.56,279.851 1642.36,280.187 \n",
" 1643.15,280.554 1643.95,280.928 1644.74,281.023 1645.54,281.195 1646.33,280.542 1647.13,280.926 1647.92,281.089 1648.71,281.58 1649.51,282.018 1650.3,282.405 \n",
" 1651.1,282.133 1651.89,280.925 1652.69,281.306 1653.48,281.648 1654.28,281.662 1655.07,281.091 1655.87,280.649 1656.66,280.89 1657.45,280.702 1658.25,280.083 \n",
" 1659.04,280.588 1659.84,280.854 1660.63,281.082 1661.43,281.021 1662.22,281.475 1663.02,281.644 1663.81,280.905 1664.61,281.295 1665.4,281.786 1666.19,281.287 \n",
" 1666.99,281.555 1667.78,281.607 1668.58,281.593 1669.37,281.728 1670.17,281.754 1670.96,281.566 1671.76,281.443 1672.55,281.335 1673.35,281.79 1674.14,281.435 \n",
" 1674.93,280.442 1675.73,280.825 1676.52,280.851 1677.32,281.305 1678.11,281.215 1678.91,280.098 1679.7,280.33 1680.5,280.375 1681.29,280.282 1682.09,280.764 \n",
" 1682.88,281.147 1683.68,281.398 1684.47,281.184 1685.26,280.391 1686.06,280.519 1686.85,280.684 1687.65,280.694 1688.44,281.054 1689.24,280.955 1690.03,281.377 \n",
" 1690.83,281.866 1691.62,282.059 1692.42,282.474 1693.21,281.905 1694,281.382 1694.8,281.739 1695.59,282.077 1696.39,282.252 1697.18,282.059 1697.98,281.981 \n",
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" 1754.39,281.502 1755.19,281.964 1755.98,282.062 1756.77,282.397 1757.57,282.618 1758.36,282.906 1759.16,283.334 1759.95,283.489 1760.75,283.852 1761.54,284.308 \n",
" 1762.34,284.265 1763.13,284.417 1763.93,284.829 1764.72,284.834 1765.51,285.28 1766.31,285.207 1767.1,284.752 1767.9,285.074 1768.69,285.527 1769.49,284.408 \n",
" 1770.28,284.87 1771.08,284.939 1771.87,285.019 1772.67,285.078 1773.46,285.115 1774.25,285.405 1775.05,285.543 1775.84,284.834 1776.64,284.899 1777.43,285.267 \n",
" 1778.23,283.594 1779.02,283.883 1779.82,284.294 1780.61,284.643 1781.41,284.748 1782.2,285.201 1782.99,285.558 1783.79,285.545 1784.58,285.999 1785.38,286.002 \n",
" 1786.17,285.973 1786.97,285.904 1787.76,286.328 1788.56,286.519 1789.35,286.951 1790.15,287.359 1790.94,287.129 1791.73,286.688 1792.53,286.603 1793.32,285.841 \n",
" 1794.12,285.886 1794.91,285.959 1795.71,286.219 1796.5,286.438 1797.3,286.564 1798.09,286.509 1798.89,286.319 1799.68,286.16 1800.47,286.619 1801.27,287.023 \n",
" 1802.06,287.423 1802.86,287.713 1803.65,287.947 1804.45,288.389 1805.24,287.859 1806.04,286.617 1806.83,286.61 1807.63,286.779 1808.42,286.955 1809.22,286.608 \n",
" 1810.01,287.02 1810.8,287.468 1811.6,287.89 1812.39,287.991 1813.19,287.998 1813.98,288.378 1814.78,288.815 1815.57,288.933 1816.37,289.107 1817.16,289.116 \n",
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},
"execution_count": 4,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"\n",
"avg_s = cumsum(x)./(1:10_000)\n",
"\n",
"plot( 1:2000,avg_s[1:2000],\n",
" title = \"sample average over observation 1 to s\",\n",
" xlabel = \"s\")"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Task 3\n",
"\n",
"(a) Code up a function `SimulateAvg(T,NSim)` which calculates `NSim` different sample averages from `NSim` different samples of length `T` where the data is drawn from $x=z-2$ where $z$ has a `Chisq(2)` distribution. \n",
"\n",
"(b) Simulate 10000 samples of length 5, and estimate the average of x in each sample. Plot a histogram over the averages, using the bins `-3:0.1:3`. Repeat for sample lengths of 25, 100 and 1000."
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"SimulateAvg (generic function with 1 method)"
]
},
"execution_count": 5,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"function SimulateAvg(T,NSim)\n",
" avg = fill(NaN,NSim)\n",
" for i = 1:NSim\n",
" x_i = rand(Chisq(2),T) .- 2\n",
" avg[i] = mean(x_i)\n",
" end\n",
" return avg\n",
"end"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {},
"outputs": [],
"source": [
"NSim = 10_000\n",
"\n",
"avg_5 = SimulateAvg(5,NSim)\n",
"avg_25 = SimulateAvg(25,NSim)\n",
"avg_100 = SimulateAvg(100,NSim)\n",
"avg_1000 = SimulateAvg(1000,NSim);"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {},
"outputs": [
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"source": [
"bins = -3:0.1:3\n",
"\n",
"p1 = histogram( avg_5,bins = bins,\n",
" normalized = true,legend=false,\n",
" title = \"Histogram of average, T=5\" )\n",
"p2 = histogram( avg_25,bins = bins,\n",
" normalized = true,legend=false,\n",
" title = \"Histogram of average, T=25\" )\n",
"p3 = histogram( avg_100,bins = bins,\n",
" normalized = true,legend=false,\n",
" title = \"Histogram of average, T=100\" )\n",
"p4 = histogram( avg_1000,bins = bins,\n",
" normalized = true,legend=false,\n",
" title = \"Histogram of average, T=1000\" )\n",
"plot(p1,p2,p3,p4,layout = (2,2),size=(800,600))"
]
},
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"source": [
"# Task 4: Central Limit Theorem\n",
"\n",
"Multiply each averages (calculated above) with the square root of the sample size (5, 25, 100, or 1000) and divide by $\\sigma$ (to get $\\sqrt{T} \\bar{x} /\\sigma$) and plot histograms again. In each subplot, add the pdf of a N$(0,1)$ variable. Recall that $\\sigma=2$.\n",
"\n",
"Notice $\\sqrt{T} \\bar{x}$ should converge to a normally distributed variable (with a zero mean since $x$ has). Dividing by $\\sigma$ should make it a standard normal."
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {},
"outputs": [
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},
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"metadata": {},
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}
],
"source": [
"σ = 2\n",
"pdfN01 = pdf(Normal(0,1),bins)\n",
"\n",
"p1 = histogram( sqrt(5)*avg_5/σ,bins = bins,\n",
" normalized = true,legend=false,\n",
" title = \"Histogram of √T average/σ, T=5\" )\n",
"plot!(bins,pdfN01,linewidth=2)\n",
"p2 = histogram( sqrt(25)*avg_25/σ,bins = bins,\n",
" normalized = true,legend=false,\n",
" title = \"Histogram of √T average/σ, T=25\" )\n",
"plot!(bins,pdfN01,linewidth=2)\n",
"p3 = histogram( sqrt(100)*avg_100/σ,bins = bins,\n",
" normalized = true,legend=false,\n",
" title = \"Histogram of √T average/σ, T=100\" )\n",
"plot!(bins,pdfN01,linewidth=2)\n",
"p4 = histogram( sqrt(1000)*avg_1000/σ,bins = bins,\n",
" normalized = true,legend=false,\n",
" title = \"Histogram of √T average/σ, T=1000\" )\n",
"plot!(bins,pdfN01,linewidth=2)\n",
"plot(p1,p2,p3,p4,layout = (2,2),size=(800,600))"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": []
}
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