3 Using Monte Carlo simulation estimate the average distance

3. Using Monte Carlo simulation, estimate the average distance between two randomly generated points in the unit square. (You can decide on the number of points that should be generated, once you reach enough many points, the value of your estimate should stop changing!)

Solution

I have used the data analysis tool pack t generate random numbers between 0 and 1. To uderstand the steps involved, watch the following youtube video:

https://www.youtube.com/watch?v=Z_b3iSEbvtA

IF you start with a seed value, you will not get a changing number everytime

I have taken about a 1000 simulations but even those show some amount of variability in the average distance.

However, one can safely say from the results of mean distances, that the required value is about 0.5.

Note : The table that I have simulated is shown here. It only has 200 simulations because chegg editor does not allow to let me have answers over 65,000 characters.

Ask if you have any doubts.

X1	Y1	X2	Y2		Distance
0.0111	0.0371	0.1653	0.5098		0.4972		Average
0.7478	0.3435	0.7537	0.0519		0.2916		0.5063
0.7083	0.1718	0.7273	0.9920		0.8204
0.5974	0.5040	0.3518	0.3975		0.2676
0.1150	0.0315	0.6784	0.2396		0.6006
0.4775	0.2173	0.4825	0.1975		0.0205
0.1280	0.7891	0.9099	0.3482		0.8977
0.7929	0.1953	0.2328	0.5222		0.6485
0.5398	0.0557	0.6239	0.6063		0.5570
0.9942	0.1368	0.3542	0.5748		0.7756
0.3704	0.8844	0.3212	0.5307		0.3571
0.4727	0.0438	0.3129	0.5382		0.5196
0.3245	0.3992	0.9150	0.5265		0.6041
0.5796	0.1632	0.4661	0.0482		0.1616
0.3737	0.0488	0.9691	0.1720		0.6080
0.7710	0.5530	0.3624	0.2911		0.4853
0.4632	0.4859	0.0607	0.1295		0.5376
0.8767	0.2738	0.1676	0.7262		0.8411
0.3440	0.0706	0.2696	0.6798		0.6137
0.4957	0.2052	0.2158	0.4335		0.3611
0.5299	0.9998	0.7906	0.8827		0.2858
0.5148	0.4871	0.6849	0.0863		0.4354
0.3797	0.0605	0.6723	0.3622		0.4203
0.7537	0.5862	0.3973	0.5492		0.3583
0.3040	0.0095	0.6801	0.7163		0.8007
0.7344	0.9873	0.7461	0.1002		0.8872
0.0971	0.5764	0.2847	0.5527		0.1891
0.5756	0.5100	0.7454	0.2789		0.2867
0.3946	0.0934	0.7752	0.1484		0.3846
0.0125	0.2498	0.2063	0.3588		0.2223
0.2548	0.7484	0.4976	0.7353		0.2432
0.5411	0.1446	0.0380	0.4450		0.5860
0.8661	0.4646	0.2177	0.4099		0.6508
0.6876	0.1660	0.6728	0.1970		0.0344
0.3790	0.4040	0.6099	0.6691		0.3516
0.4033	0.1361	0.6176	0.1763		0.2180
0.8631	0.0962	0.8700	0.5698		0.4737
0.5841	0.1684	0.1294	0.8191		0.7938
0.3887	0.1928	0.4538	0.5141		0.3279
0.7146	0.7344	0.0161	0.0335		0.9895
0.2349	0.8458	0.9403	0.6598		0.7295
0.6380	0.3552	0.5622	0.4232		0.1018
0.2203	0.7118	0.0198	0.9553		0.3154
0.4507	0.9178	0.5798	0.1338		0.7945
0.1612	0.5808	0.6074	0.7996		0.4970
0.5216	0.6161	0.5842	0.3267		0.2960
0.4528	0.6302	0.0814	0.3280		0.4788
0.6366	0.5180	0.0946	0.2760		0.5936
0.7767	0.3035	0.6460	0.3217		0.1319
0.2684	0.7563	0.0602	0.0913		0.6968
0.9327	0.8145	0.8203	0.7433		0.1330
0.9709	0.3454	0.4106	0.0656		0.6262
0.7961	0.2759	0.0512	0.8966		0.9696
0.8985	0.6225	0.7321	0.6477		0.1683
0.3992	0.4534	0.4517	0.2123		0.2468
0.4510	0.3139	0.0677	0.0398		0.4712
0.1392	0.1447	0.6642	0.6576		0.7339
0.0429	0.2269	0.8437	0.4206		0.8238
0.1092	0.1830	0.8487	0.7717		0.9452
0.9983	0.5658	0.9199	0.7924		0.2398
0.5560	0.0665	0.7964	0.3259		0.3537
0.5695	0.8733	0.7657	0.4540		0.4629
0.4618	0.2116	0.1676	0.6086		0.4941
0.0828	0.5970	0.7608	0.0986		0.8415
0.2514	0.8325	0.8554	0.3356		0.7821
0.2070	0.2810	0.6209	0.5361		0.4862
0.6241	0.4447	0.4733	0.1853		0.3000
0.3485	0.3819	0.9485	0.0408		0.6902
0.5106	0.9939	0.9691	0.9588		0.4599
0.1707	0.7107	0.9089	0.3241		0.8333
0.1554	0.6100	0.3632	0.3648		0.3215
0.2383	0.4985	0.0306	0.1324		0.4209
0.3236	0.9863	0.6115	0.4303		0.6262
0.7872	0.0887	0.1314	0.4694		0.7583
0.6328	0.3828	0.5937	0.5352		0.1574
0.6187	0.2530	0.3690	0.4440		0.3143
0.0106	0.2555	0.2269	0.0729		0.2831
0.1193	0.1328	0.4169	0.7426		0.6786
0.2760	0.5103	0.7038	0.7351		0.4832
0.4051	0.8056	0.7651	0.7274		0.3684
0.8470	0.3818	0.8556	0.4223		0.0414
0.5913	0.4765	0.1459	0.1583		0.5474
0.5723	0.9372	0.6409	0.7786		0.1728
0.1868	0.2946	0.0847	0.5412		0.2670
0.6882	0.2052	0.7584	0.3520		0.1627
0.6137	0.7943	0.5759	0.1003		0.6950
0.9012	0.9302	0.3857	0.3799		0.7541
0.8519	0.9620	0.8834	0.3758		0.5870
0.5946	0.9507	0.0420	0.1986		0.9333
0.2085	0.9269	0.4664	0.4467		0.5451
0.1603	0.2036	0.5770	0.2789		0.4234
0.2123	0.2782	0.0306	0.7781		0.5319
0.4567	0.3530	0.2837	0.8863		0.5607
0.6338	0.0059	0.7037	0.6944		0.6921
0.3896	0.0438	0.1368	0.3659		0.4094
0.6289	0.0674	0.3359	0.4766		0.5032
0.6201	0.7814	0.1564	0.0524		0.8640
0.0091	0.5790	0.9262	0.0848		1.0418
0.3964	0.4345	0.8955	0.8062		0.6222
0.6360	0.6321	0.1728	0.5064		0.4800
0.5106	0.3650	0.0539	0.1721		0.4958
0.9404	0.7348	0.1491	0.7283		0.7913
0.3810	0.1815	0.2163	0.1658		0.1655
0.5672	0.8782	0.1044	0.3345		0.7140
0.2589	0.1180	0.7156	0.6833		0.7267
0.1448	0.2279	0.3898	0.7293		0.5580
0.5610	0.0381	0.6210	0.5362		0.5017
0.1797	0.4403	0.5085	0.9843		0.6356
0.3259	0.0639	0.8515	0.1119		0.5278
0.0759	0.6041	0.2923	0.3114		0.3639
0.7966	0.8298	0.4143	0.6590		0.4188
0.2795	0.8065	0.2019	0.1608		0.6504
0.1274	0.3613	0.7676	0.1968		0.6610
0.5484	0.3249	0.7529	0.9427		0.6507
0.2157	0.5793	0.3093	0.0513		0.5362
0.3466	0.4433	0.0662	0.3759		0.2884
0.6595	0.4284	0.9903	0.0146		0.5297
0.3637	0.8933	0.5445	0.4599		0.4697
0.8388	0.6315	0.0523	0.1307		0.9324
0.1596	0.9210	0.6724	0.0728		0.9911
0.1232	0.0682	0.8918	0.1050		0.7695
0.9341	0.9779	0.9425	0.1941		0.7839
0.2038	0.7788	0.0478	0.3401		0.4657
0.4211	0.0379	0.9059	0.7518		0.8629
0.5498	0.5935	0.3152	0.5954		0.2346
0.9085	0.5384	0.3486	0.0956		0.7139
0.9912	0.3858	0.9833	0.2727		0.1134
0.3825	0.9465	0.5915	0.0944		0.8773
0.4254	0.9518	0.1989	0.3261		0.6654
0.7971	0.9507	0.9169	0.8588		0.1509
0.6493	0.5141	0.4547	0.7973		0.3436
0.4689	0.2771	0.1185	0.0889		0.3978
0.3164	0.8504	0.2024	0.9755		0.1692
0.5964	0.1313	0.1782	0.0471		0.4266
0.0188	0.0478	0.5911	0.1828		0.5881
0.9045	0.2641	0.0658	0.1554		0.8458
0.4942	0.8814	0.3306	0.1845		0.7158
0.4142	0.6633	0.6655	0.2182		0.5112
0.9579	0.8184	0.6798	0.2239		0.6564
0.3370	0.8701	0.8265	0.2703		0.7742
0.5601	0.6467	0.5594	0.6824		0.0357
0.0945	0.9204	0.5386	0.0489		0.9781
0.9676	0.7293	0.7911	0.3652		0.4046
0.4631	0.9113	0.2333	0.0919		0.8510
0.0709	0.8828	0.4601	0.4110		0.6116
0.8431	0.1924	0.2092	0.4601		0.6881
0.8163	0.8814	0.4039	0.8272		0.4160
0.6534	0.1823	0.1832	0.0851		0.4801
0.1630	0.5861	0.8250	0.6501		0.6651
0.8515	0.8101	0.9661	0.7419		0.1333
0.1650	0.6980	0.0298	0.7303		0.1390
0.5772	0.5815	0.2631	0.6465		0.3208
0.1783	0.1377	0.2917	0.1425		0.1135
0.9230	0.2705	0.5999	0.6785		0.5205
0.1925	0.0492	0.8391	0.2207		0.6690
0.8277	0.2355	0.3737	0.2291		0.4541
0.2884	0.4286	0.9690	0.2186		0.7122
0.0969	0.3613	0.0284	0.6733		0.3194
0.2198	0.3779	0.2850	0.5947		0.2264
0.5468	0.6247	0.3557	0.4678		0.2473
0.1204	0.9844	0.0610	0.9222		0.0860
0.2741	0.2865	0.9210	0.8737		0.8736
0.3358	0.8240	0.7291	0.4242		0.5608
0.0504	0.7078	0.6139	0.3039		0.6932
0.3810	0.0905	0.4928	0.1344		0.1201
0.8418	0.7910	0.3242	0.2963		0.7160
0.0218	0.3496	0.0658	0.6814		0.3348
0.4532	0.0469	0.7428	0.1107		0.2965
0.4823	0.9172	0.5349	0.6996		0.2239
0.2983	0.2857	0.2868	0.9520		0.6664
0.7767	0.8641	0.3482	0.8638		0.4285
0.2936	0.9331	0.1511	0.8177		0.1834
0.1664	0.6444	0.4287	0.5499		0.2788
0.8784	0.9725	0.4839	0.9706		0.3945
0.7421	0.3493	0.4124	0.1194		0.4020
0.1590	0.5106	0.6872	0.0364		0.7099
0.1306	0.5867	0.0114	0.8304		0.2712
0.1782	0.9694	0.8431	0.7263		0.7079
0.3270	0.9843	0.5025	0.3717		0.6373
0.3104	0.9022	0.2639	0.1847		0.7190
0.6519	0.3188	0.3406	0.0240		0.4288
0.7802	0.4353	0.1687	0.9621		0.8072
0.8332	0.2701	0.8963	0.4443		0.1852
0.3079	0.3343	0.4834	0.6124		0.3288
0.2129	0.7999	0.3198	0.0765		0.7313
0.8781	0.6953	0.7670	0.9155		0.2467
0.0797	0.1555	0.5292	0.1864		0.4506
0.6352	0.2617	0.2622	0.7256		0.5953
0.1258	0.4990	0.3233	0.5218		0.1989
0.9020	0.3653	0.0712	0.4429		0.8344
0.0285	0.1611	0.6203	0.5976		0.7353
0.3249	0.4928	0.4565	0.1122		0.4027
0.1581	0.5199	0.8209	0.6063		0.6684
0.1433	0.4773	0.2666	0.1450		0.3545
0.4091	0.5049	0.2966	0.9528		0.4617
0.5836	0.3145	0.4869	0.6670		0.3655
0.1575	0.7261	0.0031	0.4146		0.3477
0.3797	0.6044	0.9154	0.4914		0.5475
0.3432	0.2274	0.2185	0.9272		0.7109
0.4165	0.6173	0.9634	0.7160		0.5558