MonteCarlo Simulation

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Description:

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This repository illustrates how MonteCarlo calculates crucial decision-making tools from simulations $X \sim f(X)$ like E[X]:

$$E[X] \approx \frac{1}{N} \sum_{i=1}^{N} x_i = \mu_{M.C}$$

$X$ = Random Variable from simulations.
$\mu_{M.C}$ = Mean of the MonteCarlo Simulations.
$N$ = No° of Simulations.

& E[RoI] and probabilities with their density function f(X) & cumulative distribution F(X):

In this case the expected Capital per game in $(n=100)$ games planned to be played in a Casi-no is:

$$E[X_1]+ E[X_2] + ... + E[X_n] = \mu_{MC{_{1,2,...n}}}$$

Results:

The Expectancy of the Capital could have the following outcomes for $E[X_{1,2,.., n}]$:

	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30	31	32	33	34	35	36	37	38	39	40	41	42	43	44	45	46	47	48	49	50	51	52	53	54	55	56	57	58	59	60	61	62	63	64	65	66	67	68	69	70	71	72	73	74	75	76	77	78	79	80	81	82	83	84	85	86	87	88	89	90	91	92	93	94	95	96	97	98	99	100
$E[X]$	50	49	48	47	48.286	47.286	47.9132	46.9132	47.7366	46.7366	47.5672	46.5672	47.3213	46.3213	47.1186	46.1186	46.8925	45.8925	46.7285	45.7285	46.5285	45.5285	46.2754	45.2754	46.0646	45.0646	45.8826	44.8826	45.6961	44.6961	45.5798	44.5798	45.3564	44.3564	45.1555	44.1555	44.9609	43.9609	44.751	43.751	44.5789	43.5789	44.3744	43.3744	44.1681	43.1681	43.8997	42.8997	43.7483	42.7483	43.5285	42.5285	43.3519	42.3519	43.2113	42.2113	42.9933	41.9933	42.8149	41.8149	42.5978	41.5978	42.4176	41.4176	42.1798	41.1798	42.0077	41.0077	41.8041	40.8041	41.6041	40.6041	41.4005	40.4005	41.2023	40.2023	41.0032	40.0032	40.7933	39.7933	40.5699	39.5699	40.378	39.378	40.1195	39.1195	39.9519	38.9519	39.7762	38.7762	39.524	38.524	39.2979	38.2979	39.061	38.061	38.8745	37.8745	38.6664	37.6664
$E[RoI]$	0	-0.02	-0.04	-0.06	-0.03428	-0.05428	-0.041736	-0.061736	-0.045268	-0.065268	-0.048656	-0.068656	-0.053574	-0.073574	-0.057628	-0.077628	-0.06215	-0.08215	-0.06543	-0.08543	-0.06943	-0.08943	-0.074492	-0.094492	-0.078708	-0.098708	-0.082348	-0.102348	-0.086078	-0.106078	-0.088404	-0.108404	-0.092872	-0.112872	-0.09689	-0.11689	-0.100782	-0.120782	-0.10498	-0.12498	-0.108422	-0.128422	-0.112512	-0.132512	-0.116638	-0.136638	-0.122006	-0.142006	-0.125034	-0.145034	-0.12943	-0.14943	-0.132962	-0.152962	-0.135774	-0.155774	-0.140134	-0.160134	-0.143702	-0.163702	-0.148044	-0.168044	-0.151648	-0.171648	-0.156404	-0.176404	-0.159846	-0.179846	-0.163918	-0.183918	-0.167918	-0.187918	-0.17199	-0.19199	-0.175954	-0.195954	-0.179936	-0.199936	-0.184134	-0.204134	-0.188602	-0.208602	-0.19244	-0.21244	-0.19761	-0.21761	-0.200962	-0.220962	-0.204476	-0.224476	-0.20952	-0.22952	-0.214042	-0.234042	-0.21878	-0.23878	-0.22251	-0.24251	-0.226672	-0.246672

At the $100_{th}$ game the probability to Win is:

	$Pr(E[X_n] \geq 50)$
True	0.1944
False	0.8056

Probabilities are illustrated with their frequencies:

$x_i$	frequency	$f(x)$	$F(x)$
-31	3	0.0003	0.0003
-22	16	0.0016	0.0019
-13	54	0.0054	0.0073
-4	189	0.0189	0.0262
5	435	0.0435	0.0697
14	914	0.0914	0.1611
23	1422	0.1422	0.3033
32	1714	0.1714	0.4747
41	1846	0.1846	0.6593
50	1463	0.1463	0.8056
59	1021	0.1021	0.9077
68	576	0.0576	0.9653
77	227	0.0227	0.988
86	91	0.0091	0.9971
95	22	0.0022	0.9993
104	6	0.0006	0.9999
113	1	0.0001	1

Resulting $f(X)$ on the Winning Games in a $100_{th}$ played & the $\mu_{MC{_{1,2,...n}}}$ should be the same as $X$ is discrete for both games and Capital:

Note: $\left|\mu_{MC{{n_i}}} \right|$ - $|\mu{MC{{n{i+1}}}}|$ $\approx$ $\xi$ $\forall$ $\Delta_n$ simulations.
See Repo Visualization render for more details.

• MonteCarlo Methods
  
• E[X] → RoI
  
• f(X) ↔ F(X)
  
• LaTeX
  
• random.randrange