HSG-MCS-HS21_Julia/Problemsets/PS03_CLT.ipynb
2021-11-15 21:14:51 +01:00

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{
"cells": [
{
"cell_type": "code",
"execution_count": 99,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"printyellow (generic function with 1 method)"
]
},
"execution_count": 99,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"using Printf, Statistics, Random, Distributions\n",
"include(\"jlFiles/printmat.jl\")"
]
},
{
"cell_type": "code",
"execution_count": 100,
"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": 101,
"metadata": {},
"outputs": [
{
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"estimated mean and std: -0.002 1.950\n",
"\n",
"\u001b[31m\u001b[1mstrongly non-normal\u001b[22m\u001b[39m\n"
]
}
],
"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": 102,
"metadata": {},
"outputs": [
{
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438.88,867.238 439.693,869.516 440.505,869.227 441.317,872.12 442.13,874.576 442.942,876.719 \n 443.755,878.502 444.567,866.428 445.379,866.996 446.192,864.705 447.004,858.711 447.817,858.303 448.629,859.156 449.441,859.45 450.254,857.953 451.066,851.483 \n 451.879,851.69 452.691,846.98 453.504,849.529 454.316,849.232 455.128,851.82 455.941,853.001 456.753,855.714 457.566,855.144 458.378,855.746 459.19,854.338 \n 460.003,855.479 460.815,857.606 461.628,858.465 462.44,859.402 463.252,861.605 464.065,862.048 464.877,863.139 465.69,865.78 466.502,866.693 467.314,868.665 \n 468.127,866.631 468.939,868.046 469.752,870.365 470.564,870.42 471.377,872.779 472.189,874.876 473.001,874.408 473.814,875.179 474.626,877.345 475.439,879.619 \n 476.251,879.769 477.063,878.742 477.876,876.881 478.688,878.226 479.501,880.272 480.313,879.695 481.125,878.73 481.938,873.457 482.75,875.663 483.563,878.099 \n 484.375,879.814 485.187,881.105 486,879.88 486.812,880.104 487.625,879.327 488.437,881.597 489.249,881.732 490.062,883.564 490.874,884.247 491.687,886.151 \n 492.499,883.895 493.312,885.63 494.124,884.255 494.936,886.268 495.749,885.346 496.561,887.613 497.374,888.77 498.186,890.75 498.998,892.662 499.811,892.705 \n 500.623,894.287 501.436,895.561 502.248,897.121 503.06,899.036 503.873,900.924 504.685,902.265 505.498,903.508 506.31,905.526 507.122,903.859 507.935,903.611 \n 508.747,901.259 509.56,902.676 510.372,904.754 511.184,902.438 511.997,900.135 512.809,900.68 513.622,902.081 514.434,901.675 515.247,901.722 516.059,902.816 \n 516.871,904.335 517.684,903.875 518.496,904.817 519.309,905.41 520.121,904.663 520.933,906.031 521.746,906.41 522.558,908.463 523.371,904.628 524.183,905.058 \n 524.995,904.668 525.808,906.507 526.62,905.304 527.433,905.893 528.245,906.991 529.057,908.285 529.87,907.381 530.682,906.675 531.495,906.519 532.307,907.226 \n 533.12,906.424 533.932,907.716 534.744,908.073 535.557,909.301 536.369,911.281 537.182,912.774 537.994,913.919 538.806,913.539 539.619,915.315 540.431,914.839 \n 541.244,913.84 542.056,914.927 542.868,915.618 543.681,917.549 544.493,917.336 545.306,916.09 546.118,916.546 546.93,917.283 547.743,917.622 548.555,917.137 \n 549.368,914.923 550.18,914.408 550.992,912.258 551.805,913.97 552.617,915.261 553.43,916.259 554.242,917.897 555.055,918.456 555.867,916.883 556.679,917.932 \n 557.492,914.981 558.304,916.12 559.117,916.639 559.929,918.367 560.741,918.082 561.554,919.168 562.366,918.638 563.179,912.979 563.991,912.468 564.803,912.37 \n 565.616,909.627 566.428,911.315 567.241,912.693 568.053,913.022 568.865,912.649 569.678,912.024 570.49,912.632 571.303,913.928 572.115,912.998 572.927,913.166 \n 573.74,913.209 574.552,911.638 575.365,908.587 576.177,908.604 576.99,908.959 577.802,909.887 578.614,910.128 579.427,911.878 580.239,910.673 581.052,911.876 \n 581.864,911.362 582.676,913.058 583.489,914.362 584.301,915.262 585.114,915.964 585.926,917.171 586.738,917.424 587.551,917.518 588.363,918.758 589.176,919.903 \n 589.988,921.499 590.8,921.49 591.613,922.955 592.425,924.13 593.238,924.257 594.05,924.453 594.863,920.278 595.675,920.992 596.487,919.677 597.3,919.094 \n 598.112,916.386 598.925,916.584 599.737,917.185 600.549,917.05 601.362,918.22 602.174,919.271 602.987,918.451 603.799,919.949 604.611,916.142 605.424,914.122 \n 606.236,914.598 607.049,914.425 607.861,915.996 608.673,917.008 609.486,916.499 610.298,917.176 611.111,916.927 611.923,913.214 612.735,912.974 613.548,911.741 \n 614.36,913.172 615.173,912.764 615.985,913.583 616.798,914.973 617.61,912.961 618.422,914.437 619.235,914.492 620.047,916.063 620.86,916.414 621.672,917.582 \n 622.484,918.871 623.297,920.111 624.109,921.257 624.922,922.708 625.734,920.039 626.546,921.429 627.359,920.046 628.171,920.689 628.984,920.952 629.796,921.632 \n 630.608,923.134 631.421,916.409 632.233,916.614 633.046,916.209 633.858,917.026 634.671,918.499 635.483,918.765 636.295,917.563 637.108,915.434 637.92,914.397 \n 638.733,915.204 639.545,915.519 640.357,916.913 641.17,916.51 641.982,916.475 642.795,917.826 643.607,917.603 644.419,918.962 645.232,919.378 646.044,920.665 \n 646.857,920.117 647.669,919.774 648.481,917.841 649.294,918.907 650.106,914.734 650.919,915.212 651.731,914.3 652.543,915.363 653.356,914.912 654.168,916.311 \n 654.981,917.709 655.793,918.798 656.606,916.154 657.418,914.238 658.23,913.43 659.043,914.253 659.855,915.23 660.668,914.353 661.48,914.78 662.292,916.062 \n 663.105,916.621 663.917,917.508 664.73,917.642 665.542,917.16 666.354,917.954 667.167,918.539 667.979,917.936 668.792,918.204 669.604,912.677 670.416,913.246 \n 671.229,914.012 672.041,915.06 672.854,914.482 673.666,914.619 674.478,915.327 675.291,912.088 676.103,913.099 676.916,912.112 677.728,911.892 678.541,910.895 \n 679.353,909.81 680.165,910.053 680.978,910.175 681.79,911.007 682.603,912.046 683.415,912.615 684.227,912.257 685.04,912.549 685.852,913.121 686.665,912.124 \n 687.477,911.731 688.289,910.494 689.102,911.673 689.914,908.948 690.727,908.751 691.539,908.758 692.351,909.41 693.164,908.889 693.976,909.879 694.789,905.452 \n 695.601,905.528 696.414,904.969 697.226,905.308 698.038,905.945 698.851,906.661 699.663,905.012 700.476,906.068 701.288,906.17 702.1,907.452 702.913,905.74 \n 703.725,906.666 704.538,907.056 705.35,906.481 706.162,905.598 706.975,906.892 707.787,906.556 708.6,907.844 709.412,908.537 710.224,908.319 711.037,907.278 \n 711.849,907.265 712.662,904.78 713.474,904.088 714.286,905.375 715.099,906.275 715.911,906.922 716.724,905.867 717.536,905.841 718.349,905.066 719.161,905.45 \n 719.973,906.669 720.786,907.562 721.598,906.241 722.411,907.229 723.223,907.852 724.035,907.22 724.848,905.932 725.66,906.138 726.473,907.159 727.285,906.428 \n 728.097,906.396 728.91,905.512 729.722,906.302 730.535,905.919 731.347,905.965 732.159,904.893 732.972,906.057 733.784,907.19 734.597,907.604 735.409,908.835 \n 736.222,909.676 737.034,910.194 737.846,911.353 738.659,912.192 739.471,911.4 740.284,912.377 741.096,913.082 741.908,912.156 742.721,913.254 743.533,913.604 \n 744.346,914.694 745.158,914.26 745.97,914.165 746.783,914.674 747.595,915.318 748.408,913.834 749.22,914.336 750.032,915.295 750.845,914.552 751.657,913.822 \n 752.47,914.182 753.282,913.569 754.094,913.941 754.907,914.782 755.719,915.922 756.532,916.678 757.344,917.272 758.157,917.696 758.969,918.198 759.781,917.177 \n 760.594,915.917 761.406,915.656 762.219,916.138 763.031,913.939 763.843,910.562 764.656,908.815 765.468,909.058 766.281,909.116 767.093,907.959 767.905,908.716 \n 768.718,906.868 769.53,907.771 770.343,907.966 771.155,907.958 771.967,907.285 772.78,908.195 773.592,908.104 774.405,908.117 775.217,906.443 776.029,907.183 \n 776.842,907.442 777.654,907.28 778.467,907.971 779.279,908.095 780.092,908.252 780.904,908.056 781.716,906.469 782.529,906.414 783.341,907.361 784.154,907.094 \n 784.966,902.068 785.778,902.676 786.591,903.624 787.403,904.282 788.216,905.394 789.028,905.821 789.84,906.268 790.653,907.047 791.465,906.471 792.278,906.019 \n 793.09,906.391 793.902,906.574 794.715,905.845 795.527,906.391 796.34,906.208 797.152,907.241 797.965,905.505 798.777,905.785 799.589,905.206 800.402,906.26 \n 801.214,907.122 802.027,906.043 802.839,905.482 803.651,906.421 804.464,907.149 805.276,906.757 806.089,907.443 806.901,907.167 807.713,908.229 808.526,909.187 \n 809.338,910.017 810.151,910.272 810.963,907.748 811.775,908.002 812.588,909.009 813.4,907.054 814.213,907.699 815.025,908.047 815.837,908.136 816.65,909.052 \n 817.462,910.104 818.275,911.07 819.087,911.756 819.9,912.417 820.712,911.871 821.524,911.673 822.337,912.081 823.149,912.016 823.962,913.052 824.774,913.956 \n 825.586,915.013 826.399,914.871 827.211,915.052 828.024,916.01 828.836,916.884 829.648,917.065 830.461,914.695 831.273,915.092 832.086,914.623 832.898,914.415 \n 833.71,914.982 834.523,915.527 835.335,916.561 836.148,917.555 836.96,916.62 837.773,917.363 838.585,918.285 839.397,918.514 840.21,918.18 841.022,918.385 \n 841.835,918.972 842.647,919.644 843.459,918.114 844.272,917.107 845.084,913.953 845.897,913.432 846.709,914.198 847.521,915.055 848.334,915.856 849.146,915.633 \n 849.959,909.359 850.771,910.299 851.583,910.478 852.396,911.265 853.208,910.683 854.021,911.313 854.833,911.85 855.645,911.978 856.458,912.892 857.27,910.392 \n 858.083,911.089 858.895,911.545 859.708,911.283 860.52,911.486 861.332,912.045 862.145,911.015 862.957,911.887 863.77,912.691 864.582,913.16 865.394,913.329 \n 866.207,914.104 867.019,915.053 867.832,915.248 868.644,915.668 869.456,916.332 870.269,916.709 871.081,916.895 871.894,917.818 872.706,918.633 873.518,919.511 \n 874.331,920.128 875.143,920.708 875.956,921.67 876.768,922.471 877.58,922.58 878.393,922.796 879.205,922.658 880.018,923.42 880.83,923.329 881.643,923.676 \n 882.455,922.164 883.267,922.932 884.08,922.924 884.892,922.272 885.705,922.808 886.517,923.739 887.329,922.661 888.142,923.108 888.954,923.8 889.767,923.845 \n 890.579,923.178 891.391,922.899 892.204,922.204 893.016,922.732 893.829,922.555 894.641,923.445 895.453,923.828 896.266,924.22 897.078,924.996 897.891,924.909 \n 898.703,925.537 899.516,926.346 900.328,924.858 901.14,924.451 901.953,923.299 902.765,924.076 903.578,923.595 904.39,920.691 905.202,919.989 906.015,916.797 \n 906.827,917.263 907.64,917.843 908.452,918.312 909.264,914.642 910.077,915.091 910.889,915.913 911.702,916.103 912.514,916.851 913.326,917.26 914.139,916.316 \n 914.951,917.143 915.764,917.44 916.576,917.44 917.388,917.36 918.201,917.762 919.013,917.43 919.826,916.795 920.638,917.241 921.451,916.301 922.263,916.375 \n 923.075,915.384 923.888,915.521 924.7,916.358 925.513,916.746 926.325,917.3 927.137,917.688 927.95,918.453 928.762,919.159 929.575,918.196 930.387,918.991 \n 931.199,919.732 932.012,917.24 932.824,915.726 933.637,914.056 934.449,914.857 935.261,914.242 936.074,913.538 936.886,913.787 937.699,913.669 938.511,914.523 \n 939.324,915.238 940.136,914.338 940.948,915.175 941.761,915.081 942.573,913.511 943.386,912.591 944.198,913.457 945.01,914.274 945.823,914.472 946.635,909.118 \n 947.448,909.677 948.26,909.012 949.072,908.683 949.885,909.084 950.697,907.969 951.51,908.739 952.322,907.774 953.134,907.904 953.947,908.362 954.759,908.783 \n 955.572,909.219 956.384,909.961 957.196,910.24 958.009,910.264 958.821,910.62 959.634,909.087 960.446,909.826 961.259,909.641 962.071,909.11 962.883,909.746 \n 963.696,910.165 964.508,908.541 965.321,909.053 966.133,909.033 966.945,909.12 967.758,909.289 968.57,908.704 969.383,908.685 970.195,909.553 971.007,909.076 \n 971.82,909.597 972.632,909.844 973.445,910.428 974.257,908.137 975.069,908.507 975.882,908.608 976.694,909.424 977.507,909.375 978.319,909.667 979.131,909.543 \n 979.944,908.885 980.756,909.683 981.569,910.238 982.381,910.992 983.194,911.494 984.006,910.93 984.818,911.49 985.631,911.893 986.443,912.545 987.256,913.057 \n 988.068,913.076 988.88,913.557 989.693,913.87 990.505,914.342 991.318,913.075 992.13,913.054 992.942,913.454 993.755,913.934 994.567,913.181 995.38,913.119 \n 996.192,912.215 997.004,911.626 997.817,912.241 998.629,912.822 999.442,913.32 1000.25,911.77 1001.07,912.407 1001.88,913.185 1002.69,913.765 1003.5,914.028 \n 1004.32,914.751 1005.13,914.963 1005.94,914.306 1006.75,915.107 1007.57,915.223 1008.38,915.976 1009.19,914.473 1010,914.829 1010.82,915.491 1011.63,916.153 \n 1012.44,915.185 1013.25,914.852 1014.07,915.424 1014.88,914.51 1015.69,914.94 1016.5,915.134 1017.31,915.623 1018.13,914.964 1018.94,914.237 1019.75,914.926 \n 1020.56,915.711 1021.38,915.834 1022.19,915.9 1023,916.064 1023.81,916.622 1024.63,917.321 1025.44,917.9 1026.25,917.816 1027.06,915.462 1027.88,915.293 \n 1028.69,914.372 1029.5,915.084 1030.31,914.622 1031.13,915.189 1031.94,915.972 1032.75,915.653 1033.56,916.13 1034.38,916.409 1035.19,914.968 1036,914.908 \n 1036.81,915.566 1037.62,915.588 1038.44,914.014 1039.25,913.724 1040.06,912.882 1040.87,912.2 1041.69,912.313 1042.5,911.824 1043.31,912.167 1044.12,910.771 \n 1044.94,909.587 1045.75,910.321 1046.56,910.233 1047.37,909.784 1048.19,909.327 1049,909.955 1049.81,909.912 1050.62,910.362 1051.44,910.447 1052.25,909.951 \n 1053.06,909.055 1053.87,909.51 1054.69,909.383 1055.5,909.984 1056.31,909.867 1057.12,909.882 1057.94,910.628 1058.75,911.221 1059.56,909.816 1060.37,909.994 \n 1061.18,910.217 1062,910.465 1062.81,910.681 1063.62,909.662 1064.43,910.417 1065.25,910.929 1066.06,911.014 1066.87,910.589 1067.68,910.528 1068.5,911.004 \n 1069.31,910.774 1070.12,910.97 1070.93,911.668 1071.75,912.297 1072.56,912.547 1073.37,913.097 1074.18,913.652 1075,914.035 1075.81,913.07 1076.62,913.748 \n 1077.43,913.388 1078.25,913.117 1079.06,912.555 1079.87,912.808 1080.68,913.155 1081.49,913.66 1082.31,913.944 1083.12,913.74 1083.93,913.316 1084.74,913.727 \n 1085.56,913.745 1086.37,914.063 1087.18,913.967 1087.99,914.428 1088.81,915.085 1089.62,915.831 1090.43,915.821 1091.24,916.313 1092.06,916.476 1092.87,916.364 \n 1093.68,916.84 1094.49,917.541 1095.31,917.209 1096.12,916.253 1096.93,914.045 1097.74,911.682 1098.56,912.319 1099.37,912.468 1100.18,912.165 1100.99,912.658 \n 1101.81,913.197 1102.62,912.809 1103.43,913.284 1104.24,913.675 1105.05,914.299 1105.87,914.951 1106.68,914.034 1107.49,914.715 1108.3,915.278 1109.12,915.039 \n 1109.93,915.385 1110.74,914.844 1111.55,914.597 1112.37,914.749 1113.18,914.154 1113.99,914.331 1114.8,914.57 1115.62,915.177 1116.43,915.551 1117.24,915.594 \n 1118.05,914.867 1118.87,913.743 1119.68,913.395 1120.49,914.023 1121.3,914.411 1122.12,914.974 1122.93,915.325 1123.74,913.9 1124.55,914.003 1125.36,914.648 \n 1126.18,914.792 1126.99,915.24 1127.8,915.265 1128.61,915.237 1129.43,915.606 1130.24,915.02 1131.05,914.632 1131.86,914.351 1132.68,914.215 1133.49,914.238 \n 1134.3,914.225 1135.11,912.49 1135.93,912.42 1136.74,912.675 1137.55,913.315 1138.36,913.664 1139.18,913.43 1139.99,913.941 1140.8,914.496 1141.61,915.143 \n 1142.43,915.363 1143.24,915.062 1144.05,915.575 1144.86,915.749 1145.68,915.962 1146.49,914.257 1147.3,914.643 1148.11,915.279 1148.92,915.733 1149.74,916.14 \n 1150.55,916.706 1151.36,917.183 1152.17,916.732 1152.99,916.811 1153.8,917.061 1154.61,916.654 1155.42,917.048 1156.24,917.238 1157.05,916.806 1157.86,915.142 \n 1158.67,915.753 1159.49,915.972 1160.3,915.149 1161.11,915.539 1161.92,915.589 1162.74,916.226 1163.55,913.596 1164.36,910.748 1165.17,910.598 1165.99,910.556 \n 1166.8,911.208 1167.61,911.112 1168.42,911.783 1169.23,911.371 1170.05,911.608 1170.86,911.313 1171.67,911.69 1172.48,911.661 1173.3,912.163 1174.11,912.365 \n 1174.92,912.865 1175.73,912.435 1176.55,911.426 1177.36,911.723 1178.17,912.015 1178.98,912.335 1179.8,912.958 1180.61,912.11 1181.42,911.6 1182.23,911.075 \n 1183.05,911.492 1183.86,911.997 1184.67,911.105 1185.48,911.688 1186.3,911.948 1187.11,912.452 1187.92,910.371 1188.73,910.787 1189.55,910.103 1190.36,910.082 \n 1191.17,909.685 1191.98,909.332 1192.79,909.678 1193.61,908.053 1194.42,907.73 1195.23,908.364 1196.04,908.739 1196.86,908.056 1197.67,908.55 1198.48,908.895 \n 1199.29,909.301 1200.11,909.221 1200.92,909.699 1201.73,910.244 1202.54,909.75 1203.36,910.28 1204.17,910.678 1204.98,910.838 1205.79,911.077 1206.61,910.696 \n 1207.42,908.374 1208.23,908.914 1209.04,909.526 1209.86,908.455 1210.67,908.829 1211.48,908.639 1212.29,909.166 1213.1,909.462 1213.92,910.01 1214.73,909.592 \n 1215.54,910.055 1216.35,910.029 1217.17,910.234 1217.98,910.258 1218.79,910.915 1219.6,911.474 1220.42,911.615 1221.23,911.16 1222.04,910.955 1222.85,910.381 \n 1223.67,911.004 1224.48,911.343 1225.29,911.237 1226.1,910.555 1226.92,910.812 1227.73,910.349 1228.54,910.66 1229.35,910.785 1230.17,911.143 1230.98,911.761 \n 1231.79,912.022 1232.6,911.663 1233.42,909.05 1234.23,909.342 1235.04,909.722 1235.85,909.756 1236.66,907.646 1237.48,907.979 1238.29,908.333 1239.1,908.852 \n 1239.91,909.097 1240.73,908.576 1241.54,909.064 1242.35,908.842 1243.16,908.96 1243.98,908.819 1244.79,909.268 1245.6,908.422 1246.41,908.545 1247.23,908.672 \n 1248.04,909.241 1248.85,909.789 1249.66,909.828 1250.48,910 1251.29,909.818 1252.1,910.405 1252.91,910.835 1253.73,911.297 1254.54,911.072 1255.35,911.076 \n 1256.16,911.472 1256.98,911.733 1257.79,912.006 1258.6,912.524 1259.41,912.513 1260.22,910.334 1261.04,910.737 1261.85,909.812 1262.66,910.419 1263.47,911.019 \n 1264.29,911.551 1265.1,911.446 1265.91,910.669 1266.72,910.704 1267.54,910.955 1268.35,910.949 1269.16,910.695 1269.97,910.488 1270.79,910.986 1271.6,910.894 \n 1272.41,910.547 1273.22,910.407 1274.04,910.932 1274.85,910.712 1275.66,910.521 1276.47,910.024 1277.29,909.035 1278.1,909.63 1278.91,908.604 1279.72,908.708 \n 1280.53,909.111 1281.35,909.31 1282.16,909.556 1282.97,909.724 1283.78,909.712 1284.6,910.028 1285.41,909.921 1286.22,909.841 1287.03,910.216 1287.85,910.571 \n 1288.66,910.528 1289.47,911.089 1290.28,911.622 1291.1,912.01 1291.91,911.683 1292.72,911.914 1293.53,912.481 1294.35,912.988 1295.16,912.721 1295.97,912.868 \n 1296.78,913.386 1297.6,913.968 1298.41,914.122 1299.22,914.454 1300.03,914.32 1300.85,913.43 1301.66,913.9 1302.47,914.41 1303.28,914.672 1304.09,914.507 \n 1304.91,914.724 1305.72,915.216 1306.53,914.026 1307.34,914.612 1308.16,915.085 1308.97,913.906 1309.78,913.847 1310.59,914.125 1311.41,914.65 1312.22,914.661 \n 1313.03,915.225 1313.84,915.166 1314.66,915.618 1315.47,915.621 1316.28,915.246 1317.09,914.596 1317.91,914.59 1318.72,914.444 1319.53,915.004 1320.34,912.448 \n 1321.16,913.039 1321.97,913.004 1322.78,913.015 1323.59,912.28 1324.4,911.348 1325.22,911.647 1326.03,911.264 1326.84,911.642 1327.65,911.601 1328.47,910.624 \n 1329.28,910.625 1330.09,910.939 1330.9,911.019 1331.72,911.416 1332.53,911.246 1333.34,911.435 1334.15,911.959 1334.97,912.501 1335.78,913.034 1336.59,913.11 \n 1337.4,913.182 1338.22,913.7 1339.03,914.224 1339.84,914.14 1340.65,914.196 1341.47,914.498 1342.28,914.965 1343.09,914.722 1343.9,914.649 1344.72,915.136 \n 1345.53,915.111 1346.34,915.139 1347.15,914.152 1347.96,914.309 1348.78,913.674 1349.59,914.049 1350.4,914.345 1351.21,914.009 1352.03,914.339 1352.84,914.847 \n 1353.65,914.047 1354.46,914.545 1355.28,914.845 1356.09,915.376 1356.9,915.747 1357.71,916.151 1358.53,916.283 1359.34,916.213 1360.15,916.696 1360.96,917.224 \n 1361.78,917.118 1362.59,914.819 1363.4,914.835 1364.21,914.028 1365.03,914.225 1365.84,914.762 1366.65,915.166 1367.46,915.403 1368.27,915.918 1369.09,916.395 \n 1369.9,916.442 1370.71,916.62 1371.52,916.624 1372.34,917.12 1373.15,916.763 1373.96,915.392 1374.77,915.836 1375.59,915.214 1376.4,915.52 1377.21,915.356 \n 1378.02,915.556 1378.84,915.346 1379.65,915.584 1380.46,915.113 1381.27,915.621 1382.09,915.48 1382.9,915.61 1383.71,914.25 1384.52,914.729 1385.34,914.557 \n 1386.15,914.619 1386.96,913.593 1387.77,913.765 1388.59,913.473 1389.4,913.597 1390.21,913.905 1391.02,913.847 1391.83,913.86 1392.65,913.134 1393.46,913.407 \n 1394.27,913.438 1395.08,913.455 1395.9,913.96 1396.71,914.395 1397.52,914.357 1398.33,914.522 1399.15,914.949 1399.96,914.731 1400.77,915.191 1401.58,915.081 \n 1402.4,915.116 1403.21,915.209 1404.02,915.595 1404.83,915.802 1405.65,915.609 1406.46,915.929 1407.27,916.409 1408.08,916.643 1408.9,916.931 1409.71,916.813 \n 1410.52,917.335 1411.33,917.534 1412.14,917.903 1412.96,918.153 1413.77,916.296 1414.58,916.53 1415.39,917.046 1416.21,917.013 1417.02,916.053 1417.83,916.303 \n 1418.64,916.772 1419.46,917.071 1420.27,914.561 1421.08,914.834 1421.89,914.909 1422.71,915.131 1423.52,915.653 1424.33,916.166 1425.14,915.833 1425.96,916.263 \n 1426.77,916.637 1427.58,914.356 1428.39,914.616 1429.21,914.975 1430.02,915.494 1430.83,915.855 1431.64,916.047 1432.46,916.024 1433.27,914.09 1434.08,914.534 \n 1434.89,914.478 1435.7,914.606 1436.52,915.05 1437.33,915.517 1438.14,916.038 1438.95,915.901 1439.77,916.027 1440.58,914.004 1441.39,914.049 1442.2,914.488 \n 1443.02,914.767 1443.83,915.232 1444.64,915.693 1445.45,915.76 1446.27,916.06 1447.08,915.788 1447.89,916.13 1448.7,915.399 1449.52,915.367 1450.33,915.694 \n 1451.14,915.194 1451.95,915.646 1452.77,915.969 1453.58,916.046 1454.39,916.454 1455.2,916.963 1456.01,917.16 1456.83,917.204 1457.64,917.54 1458.45,917.24 \n 1459.26,917.284 1460.08,917.516 1460.89,917.368 1461.7,917.889 1462.51,918.015 1463.33,918.246 1464.14,918.741 1464.95,918.653 1465.76,918.966 1466.58,918.177 \n 1467.39,918.432 1468.2,918.572 1469.01,917.58 1469.83,917.648 1470.64,918.109 1471.45,916.287 1472.26,916.039 1473.08,915.82 1473.89,915.678 1474.7,916.048 \n 1475.51,915.854 1476.33,914.943 1477.14,915.369 1477.95,915.79 1478.76,916.289 1479.57,916.538 1480.39,916.641 1481.2,916.801 1482.01,916.958 1482.82,917.144 \n 1483.64,917.522 1484.45,917.799 1485.26,917.311 1486.07,917.767 1486.89,917.408 1487.7,917.879 1488.51,918.238 1489.32,918.649 1490.14,919.006 1490.95,919.432 \n 1491.76,919.805 1492.57,920.012 1493.39,920.433 1494.2,920.817 1495.01,921.195 1495.82,920.658 1496.64,920.977 1497.45,920.512 1498.26,920.762 1499.07,921.204 \n 1499.89,921.435 1500.7,921.112 1501.51,921.273 1502.32,920.94 1503.13,920.828 1503.95,921.329 1504.76,921.173 1505.57,921.554 1506.38,922.033 1507.2,921.963 \n 1508.01,920.169 1508.82,920.179 1509.63,920.366 1510.45,920.493 1511.26,920.784 1512.07,921.04 1512.88,921.032 1513.7,921.498 1514.51,921.874 1515.32,921.531 \n 1516.13,921.946 1516.95,921.01 1517.76,919.981 1518.57,918.956 1519.38,918.845 1520.2,919.303 1521.01,919.738 1521.82,919.679 1522.63,920.076 1523.44,920.331 \n 1524.26,920.312 1525.07,920.66 1525.88,921.097 1526.69,921.323 1527.51,921.419 1528.32,921.551 1529.13,922.018 1529.94,922.199 1530.76,922.047 1531.57,922.446 \n 1532.38,922.7 1533.19,922.2 1534.01,922.151 1534.82,921.704 1535.63,922.133 1536.44,922.578 1537.26,922.787 1538.07,923.149 1538.88,922.414 1539.69,922.875 \n 1540.51,922.059 1541.32,921.46 1542.13,921.931 1542.94,922.344 1543.76,920.954 1544.57,921.157 1545.38,921.442 1546.19,921.516 1547,920.842 1547.82,921.002 \n 1548.63,921.008 1549.44,921.283 1550.25,921.337 1551.07,921.027 1551.88,919.647 1552.69,919.681 1553.5,918.854 1554.32,918.965 1555.13,919.265 1555.94,919.72 \n 1556.75,920.142 1557.57,920.069 1558.38,920.531 1559.19,921.005 1560,920.875 1560.82,920.887 1561.63,921.337 1562.44,921.697 1563.25,922.012 1564.07,921.829 \n 1564.88,921.933 1565.69,920.218 1566.5,920.68 1567.31,921.083 1568.13,921.417 1568.94,921.27 1569.75,921.231 1570.56,921.678 1571.38,921.956 1572.19,922.175 \n 1573,922.575 1573.81,922.581 1574.63,922.125 1575.44,922.264 1576.25,922.472 1577.06,922.832 1577.88,922.829 1578.69,922.9 1579.5,923.107 1580.31,923.346 \n 1581.13,923.587 1581.94,923.812 1582.75,923.474 1583.56,922.627 1584.38,922.377 1585.19,922.568 1586,921.909 1586.81,922.176 1587.63,922.082 1588.44,922.33 \n 1589.25,922.737 1590.06,923.12 1590.87,923.353 1591.69,923.768 1592.5,923.016 1593.31,922.472 1594.12,922.205 1594.94,922.162 1595.75,921.511 1596.56,921.747 \n 1597.37,922.074 1598.19,922.182 1599,922.642 1599.81,922.726 1600.62,922.523 1601.44,922.667 1602.25,922.912 1603.06,923.141 1603.87,923.172 1604.69,923.479 \n 1605.5,923.86 1606.31,924.102 1607.12,924.35 1607.94,924.184 1608.75,924.332 1609.56,924.617 1610.37,924.942 1611.18,924.705 1612,925.019 1612.81,925.39 \n 1613.62,925.187 1614.43,924.972 1615.25,925.183 1616.06,925.555 1616.87,925.624 1617.68,925.946 1618.5,926.07 1619.31,926.448 1620.12,925.854 1620.93,926.154 \n 1621.75,926.4 1622.56,924.404 1623.37,924.686 1624.18,925.032 1625,925.007 1625.81,925.462 1626.62,925.709 1627.43,925.447 1628.25,925.766 1629.06,926.103 \n 1629.87,926.48 1630.68,925.858 1631.5,925.159 1632.31,925.181 1633.12,925.552 1633.93,925.106 1634.74,924.264 1635.56,924.382 1636.37,924.625 1637.18,924.808 \n 1637.99,925.008 1638.81,925.341 1639.62,925.546 1640.43,925.336 1641.24,924.318 1642.06,924.699 1642.87,925.144 1643.68,925.402 1644.49,925.76 1645.31,925.864 \n 1646.12,924.572 1646.93,924.778 1647.74,924.606 1648.56,924.827 1649.37,924.613 1650.18,924.55 1650.99,923.852 1651.81,924.217 1652.62,924.332 1653.43,924.016 \n 1654.24,921.696 1655.05,922.046 1655.87,920.891 1656.68,921.137 1657.49,921.177 1658.3,921.586 1659.12,921.993 1659.93,922.257 1660.74,921.918 1661.55,922.342 \n 1662.37,922.514 1663.18,922.712 1663.99,922.431 1664.8,921.941 1665.62,921.018 1666.43,921.256 1667.24,920.888 1668.05,921.078 1668.87,921.527 1669.68,921.851 \n 1670.49,922.212 1671.3,922.507 1672.12,922.936 1672.93,923.302 1673.74,923.316 1674.55,923.474 1675.37,921.869 1676.18,922.3 1676.99,922.612 1677.8,922.854 \n 1678.61,922.47 1679.43,922.731 1680.24,922.893 1681.05,922.095 1681.86,922.114 1682.68,922.554 1683.49,922.474 1684.3,922.758 1685.11,922.354 1685.93,922.62 \n 1686.74,922.451 1687.55,922.555 1688.36,922.497 1689.18,922.914 1689.99,921.285 1690.8,919.88 1691.61,920.192 1692.43,920.336 1693.24,920.14 1694.05,920.238 \n 1694.86,920.252 1695.68,920.499 1696.49,919.684 1697.3,919.317 1698.11,917.956 1698.92,918.1 1699.74,917.4 1700.55,917.754 1701.36,918.156 1702.17,918.522 \n 1702.99,918.22 1703.8,917.767 1704.61,918.101 1705.42,918.424 1706.24,918.816 1707.05,918.336 1707.86,918.663 1708.67,918.523 1709.49,918.784 1710.3,919.103 \n 1711.11,919.431 1711.92,919.83 1712.74,919.58 1713.55,919.976 1714.36,920.324 1715.17,920.681 1715.99,920.875 1716.8,919.394 1717.61,918.753 1718.42,918.762 \n 1719.24,918.523 1720.05,918.905 1720.86,919.24 1721.67,919.613 1722.48,919.624 1723.3,919.683 1724.11,920.034 1724.92,919.933 1725.73,920.047 1726.55,920.082 \n 1727.36,920.377 1728.17,920.692 1728.98,921.09 1729.8,920.627 1730.61,920.918 1731.42,921.013 1732.23,921.239 1733.05,921.569 1733.86,921.571 1734.67,921.889 \n 1735.48,921.914 1736.3,922.115 1737.11,921.167 1737.92,920.386 1738.73,920.299 1739.55,919.72 1740.36,919.64 1741.17,919.953 1741.98,920.238 1742.79,920.399 \n 1743.61,920.727 1744.42,920.834 1745.23,921.013 1746.04,921.299 1746.86,920.09 1747.67,920.26 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1798.85,920.485 1799.66,920.832 \n 1800.48,920.502 1801.29,920.616 1802.1,920.945 1802.91,920.465 1803.73,920.835 1804.54,921.017 1805.35,921.263 1806.16,921.084 1806.98,921.32 1807.79,921.074 \n 1808.6,921.244 1809.41,921.022 1810.22,921.358 1811.04,921.221 1811.85,921.156 1812.66,921.402 1813.47,921.634 1814.29,921.432 1815.1,921.043 1815.91,921.43 \n 1816.72,920.605 1817.54,920.801 1818.35,920.966 1819.16,921.236 1819.97,921.322 1820.79,921.509 1821.6,920.915 1822.41,921.274 1823.22,921.492 1824.04,920.445 \n \n \"/>\n<path clip-path=\"url(#clip440)\" d=\"\nM1571.67 259.916 L1815.37 259.916 L1815.37 156.236 L1571.67 156.236 Z\n \" fill=\"#ffffff\" fill-rule=\"evenodd\" fill-opacity=\"1\"/>\n<polyline clip-path=\"url(#clip440)\" style=\"stroke:#000000; stroke-linecap:butt; stroke-linejoin:round; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 1571.67,259.916 1815.37,259.916 1815.37,156.236 1571.67,156.236 1571.67,259.916 \n \"/>\n<polyline clip-path=\"url(#clip440)\" 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},
"execution_count": 102,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"avg_s = cumsum(x)./(1:10_000)\n",
"\n",
"plot(1:2000, avg_s[1:2000], xlabel=\"s\", title=\"sample average over observation 1 to 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": 103,
"metadata": {},
"outputs": [
{
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},
"execution_count": 103,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"function SimulateAvg(T, NSim)\n",
" # Create an array with NSim elements and initialize it with the value NaN. This is bc of if we don't put a number in it later, we will receive an error on the next computation\n",
" avg = fill(NaN, NSim)\n",
" for i = 1:NSim\n",
" x_1 = rand(Chisq(2),T) .- 2\n",
" avg[i] = mean(x_1)\n",
" end\n",
" \n",
" return avg\n",
"end\n",
"\n",
"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)\n",
"\n",
"# printmat(avg_5)\n",
"\n",
"function getHistogram(values)\n",
" return histogram(values, \n",
" bins = -3:0.1:3,\n",
" normalized = true, legend=false )\n",
"end\n",
"\n",
"plot(getHistogram(avg_5), getHistogram(avg_25), getHistogram(avg_100), getHistogram(avg_1000), layout = (2,2), size=(800, 600))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"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": 115,
"metadata": {},
"outputs": [
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"source": [
"σ = 2\n",
"pdfN01 = pdf(Normal(0,1), -3:0.1:3)\n",
"\n",
"function getHistogramWithPdf(values)\n",
" p = histogram(values, \n",
" bins = -3:0.1:3,\n",
" normalized = true,\n",
" legend=false \n",
" )\n",
" plot!(-3:0.1:3, pdfN01, linewidth=2)\n",
" return p\n",
"end\n",
"\n",
"plot(getHistogramWithPdf(avg_5 * sqrt(5) / σ),\n",
" getHistogramWithPdf(avg_25 * sqrt(25) / σ),\n",
" getHistogramWithPdf(avg_100 * sqrt(100) / σ),\n",
" getHistogramWithPdf(avg_1000 * sqrt(1000) / σ),\n",
" layout = (2,2),\n",
" size = (800,600)\n",
" )"
]
}
],
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