The Distribution of Shagai /Presentation/ - Шагайны тархал

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!