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The Distribution of Shagai /Presentation/ - Шагайны тархалт

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Bat-Ochir
Bat-Ochir

This document discusses analyzing the distribution of outcomes from tossing a shagai, or Mongolian knucklebones. It presents results from 200 trials of tossing a shagai and calculates the probability of each of the four possible outcomes. It then discusses modeling the distribution theoretically and uses two methods, method of moments and maximum likelihood, to estimate the parameters of the theoretical distribution. Finally, it discusses calculating a confidence interval for the probability of one of the outcomes based on 50 trials of 20 tosses each.

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Implemented by: German-Mongolian Institute for Resources and Technology − www.gmit.edu.mn DISTRIBUTION OF SHAGAI SOPHOMORES - TEAM II BAT-OCHIR.B INDRA.B NARANBILEG.I BATTSENGEL.B ENKHNOMIN.E SUPERVISED BY: Prof. Altangerel Mr. Dorjsundui 2018/12/06
Implemented by: AGENDA 1. TRIAL OF TOSSING SHAGAI 2. GRAPHICAL ILLUSTRATION 3. THEORETICAL ASPECT 4. PARAMETER ESTIMATION  METHOD OF MOMENT  MAXIMUM LIKELIHOOD 6. CONFIDENCE INTERVAL 7. CONCLUSION German-Mongolian Institute for Resources and Technology − www.gmit.edu.mn
Implemented by: TRIAL OF TOSSING SHAGAI German-Mongolian Institute for Resources and Technology − www.gmit.edu.mn FOUR OUTCOMES Sheep (s) Goat (g) Camel (c) Horse (h)
Implemented by: German-Mongolian Institute for Resources and Technology − www.gmit.edu.mn • 200 times TRIAL OF TOSSING SHAGAI
Implemented by: GRAPHICAL ILLUSTRATION German-Mongolian Institute for Resources and Technology − www.gmit.edu.mn 38% 38% 10% 14% DISTRIBUTION OF THROWING "SHAGAI" (200) Sheep Goat Camel Horse 0.385 0.385 0.095 0.135 SHEEP GOAT CAMEL HORSE DISTRIBUTION OF THROWING "SHAGAI" (200)
Implemented by: THEORETICAL ASPECT German-Mongolian Institute for Resources and Technology − www.gmit.edu.mn X 0 1 2 3 P(X) a b c 1-a-b-c Then, the probability distribution is: If we consider • Sheep => x=0 • Goat => x=1 • Camel => x=2 • Horse => x=3
Implemented by: METHOD OF MOMENT German-Mongolian Institute for Resources and Technology − www.gmit.edu.mn X 0 1 2 3 P(X) a b c 1-a-b-c Population Sample 𝐸 𝑥 = 𝑥 ∙ 𝑃(𝑥) 𝜇1 = 1 𝑛 𝑥𝑖 𝐸 𝑥2 = 𝑥2 ∙ 𝑃(𝑥) 𝜇2 = 1 𝑛 𝑥𝑖 2 𝐸 𝑥3 = 𝑥3 ∙ 𝑃(𝑥) 𝜇3 = 1 𝑛 𝑥𝑖 3  3 − 3𝑎 − 2𝑏 − 𝑐 = 1 𝑛 𝑥𝑖 = 0.98  9 − 9𝑎 − 8𝑏 − 5𝑐 = 1 𝑛 𝑥𝑖 2 == 1.98  27 − 27𝑎 − 26𝑏 − 19𝑐 = 1 𝑛 𝑥𝑖 3 = 4.79 𝑎 = 0.385 𝑏 = 0.385 𝑐 = 0.095 1 − 𝑎 − 𝑏 − 𝑐 = 0.135 DATA 1 𝑛 𝑥𝑖 0.98 1 𝑛 𝑥𝑖 2 1.98 1 𝑛 𝑥𝑖 3 4.79
Implemented by: MAXIMUM LIKELIHOOD German-Mongolian Institute for Resources and Technology − www.gmit.edu.mn 𝐿 𝑎, 𝑏, 𝑐 = 𝑘=𝑖 𝑛 𝑓 𝑥 = 𝑎77 𝑏77 𝑐19 1 − 𝑎 − 𝑏 − 𝑐 27 𝑙𝑛𝐿 = 77 𝑙𝑛𝑎 + 77 𝑙𝑛𝑏 + 19 𝑙𝑛𝑐 + 27𝑙𝑛(1 − 𝑎 − 𝑏 − 𝑐) 𝑑𝑙𝑛𝐿 𝑑𝑎 = 77 𝑎 − 27 1 − 𝑎 − 𝑏 − 𝑐 = 0 104𝑎 = 77 − 77𝑏 − 77𝑐 𝑑𝑙𝑛𝐿 𝑑𝑏 = 77 𝑏 − 27 1 − 𝑎 − 𝑏 − 𝑐 = 0 104𝑏 = 77 − 77𝑎 − 77𝑐 𝑑𝑙𝑛𝐿 𝑑𝑐 = 19 𝑐 − 27 1 − 𝑎 − 𝑏 − 𝑐 = 0 46𝑐 = 19 − 19𝑎 − 19𝑏 𝑎 = 0.385 𝑏 = 0.385 𝑐 = 0.095 1 − 𝑎 − 𝑏 − 𝑐 = 0.135 X 0 1 2 3 P(X) a b c 1-a-b-c Frequency Relative Frequency Percent Sheep 77 0.385 39% Goat 77 0.385 39% Camel 19 0.095 10% Horse 27 0.135 14%
Implemented by: In second trial, Shagai is tossed 20 times and the probability of each outcome is calculated. This trial is conducted 50 times. FOR LARGE SAMPLE (n>30): German-Mongolian Institute for Resources and Technology − www.gmit.edu.mn CONFIDENCE INTERVAL
Implemented by: Confidence interval for the average probability of rolling a sheep: Trial number=n=50 Z-score =𝑧 𝛼/2= 1.96 (95%) [1.65 for 90%, 2.58 for 99%] Sample mean = 𝑥= 0.4100 Sample standard deviation = s = 0.095298 (0.41)±(1.96) × (0.095298)/( 50) = 0.4100 ± 0.0264 0.3836 ≤ 𝜇 ≤ 0.4364 German-Mongolian Institute for Resources and Technology − www.gmit.edu.mn CONFIDENCE INTERVAL DATA
Implemented by: German-Mongolian Institute for Resources and Technology − www.gmit.edu.mn CONCLUSION DISTRIBUTION OF SHAGAI: • DISCRETE • ONLY 4 OUTCOMES • BIASED • THREE PARAMETERS X 0 1 2 3 P(X) a b c 1-a-b-c
Implemented by: German-Mongolian Institute for Resources and Technology − www.gmit.edu.mn THANK YOU FOR THE ATTENTION!

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The Distribution of Shagai /Presentation/ - Шагайны тархалт

  • 1. Implemented by: German-Mongolian Institute for Resources and Technology − www.gmit.edu.mn DISTRIBUTION OF SHAGAI SOPHOMORES - TEAM II BAT-OCHIR.B INDRA.B NARANBILEG.I BATTSENGEL.B ENKHNOMIN.E SUPERVISED BY: Prof. Altangerel Mr. Dorjsundui 2018/12/06
  • 2. Implemented by: AGENDA 1. TRIAL OF TOSSING SHAGAI 2. GRAPHICAL ILLUSTRATION 3. THEORETICAL ASPECT 4. PARAMETER ESTIMATION  METHOD OF MOMENT  MAXIMUM LIKELIHOOD 6. CONFIDENCE INTERVAL 7. CONCLUSION German-Mongolian Institute for Resources and Technology − www.gmit.edu.mn
  • 3. Implemented by: TRIAL OF TOSSING SHAGAI German-Mongolian Institute for Resources and Technology − www.gmit.edu.mn FOUR OUTCOMES Sheep (s) Goat (g) Camel (c) Horse (h)
  • 4. Implemented by: German-Mongolian Institute for Resources and Technology − www.gmit.edu.mn • 200 times TRIAL OF TOSSING SHAGAI
  • 5. Implemented by: GRAPHICAL ILLUSTRATION German-Mongolian Institute for Resources and Technology − www.gmit.edu.mn 38% 38% 10% 14% DISTRIBUTION OF THROWING "SHAGAI" (200) Sheep Goat Camel Horse 0.385 0.385 0.095 0.135 SHEEP GOAT CAMEL HORSE DISTRIBUTION OF THROWING "SHAGAI" (200)
  • 6. Implemented by: THEORETICAL ASPECT German-Mongolian Institute for Resources and Technology − www.gmit.edu.mn X 0 1 2 3 P(X) a b c 1-a-b-c Then, the probability distribution is: If we consider • Sheep => x=0 • Goat => x=1 • Camel => x=2 • Horse => x=3
  • 7. Implemented by: METHOD OF MOMENT German-Mongolian Institute for Resources and Technology − www.gmit.edu.mn X 0 1 2 3 P(X) a b c 1-a-b-c Population Sample 𝐸 𝑥 = 𝑥 ∙ 𝑃(𝑥) 𝜇1 = 1 𝑛 𝑥𝑖 𝐸 𝑥2 = 𝑥2 ∙ 𝑃(𝑥) 𝜇2 = 1 𝑛 𝑥𝑖 2 𝐸 𝑥3 = 𝑥3 ∙ 𝑃(𝑥) 𝜇3 = 1 𝑛 𝑥𝑖 3  3 − 3𝑎 − 2𝑏 − 𝑐 = 1 𝑛 𝑥𝑖 = 0.98  9 − 9𝑎 − 8𝑏 − 5𝑐 = 1 𝑛 𝑥𝑖 2 == 1.98  27 − 27𝑎 − 26𝑏 − 19𝑐 = 1 𝑛 𝑥𝑖 3 = 4.79 𝑎 = 0.385 𝑏 = 0.385 𝑐 = 0.095 1 − 𝑎 − 𝑏 − 𝑐 = 0.135 DATA 1 𝑛 𝑥𝑖 0.98 1 𝑛 𝑥𝑖 2 1.98 1 𝑛 𝑥𝑖 3 4.79
  • 8. Implemented by: MAXIMUM LIKELIHOOD German-Mongolian Institute for Resources and Technology − www.gmit.edu.mn 𝐿 𝑎, 𝑏, 𝑐 = 𝑘=𝑖 𝑛 𝑓 𝑥 = 𝑎77 𝑏77 𝑐19 1 − 𝑎 − 𝑏 − 𝑐 27 𝑙𝑛𝐿 = 77 𝑙𝑛𝑎 + 77 𝑙𝑛𝑏 + 19 𝑙𝑛𝑐 + 27𝑙𝑛(1 − 𝑎 − 𝑏 − 𝑐) 𝑑𝑙𝑛𝐿 𝑑𝑎 = 77 𝑎 − 27 1 − 𝑎 − 𝑏 − 𝑐 = 0 104𝑎 = 77 − 77𝑏 − 77𝑐 𝑑𝑙𝑛𝐿 𝑑𝑏 = 77 𝑏 − 27 1 − 𝑎 − 𝑏 − 𝑐 = 0 104𝑏 = 77 − 77𝑎 − 77𝑐 𝑑𝑙𝑛𝐿 𝑑𝑐 = 19 𝑐 − 27 1 − 𝑎 − 𝑏 − 𝑐 = 0 46𝑐 = 19 − 19𝑎 − 19𝑏 𝑎 = 0.385 𝑏 = 0.385 𝑐 = 0.095 1 − 𝑎 − 𝑏 − 𝑐 = 0.135 X 0 1 2 3 P(X) a b c 1-a-b-c Frequency Relative Frequency Percent Sheep 77 0.385 39% Goat 77 0.385 39% Camel 19 0.095 10% Horse 27 0.135 14%
  • 9. Implemented by: In second trial, Shagai is tossed 20 times and the probability of each outcome is calculated. This trial is conducted 50 times. FOR LARGE SAMPLE (n>30): German-Mongolian Institute for Resources and Technology − www.gmit.edu.mn CONFIDENCE INTERVAL
  • 10. Implemented by: Confidence interval for the average probability of rolling a sheep: Trial number=n=50 Z-score =𝑧 𝛼/2= 1.96 (95%) [1.65 for 90%, 2.58 for 99%] Sample mean = 𝑥= 0.4100 Sample standard deviation = s = 0.095298 (0.41)±(1.96) × (0.095298)/( 50) = 0.4100 ± 0.0264 0.3836 ≤ 𝜇 ≤ 0.4364 German-Mongolian Institute for Resources and Technology − www.gmit.edu.mn CONFIDENCE INTERVAL DATA
  • 11. Implemented by: German-Mongolian Institute for Resources and Technology − www.gmit.edu.mn CONCLUSION DISTRIBUTION OF SHAGAI: • DISCRETE • ONLY 4 OUTCOMES • BIASED • THREE PARAMETERS X 0 1 2 3 P(X) a b c 1-a-b-c
  • 12. Implemented by: German-Mongolian Institute for Resources and Technology − www.gmit.edu.mn THANK YOU FOR THE ATTENTION!