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8/11/2019 Experiments Another Group
1/24
Bernoulli Distribution - Bern(0.5)
1 Flip a coin and record the outcome. Trial Outcome
2 Compute the (population) mean and variance 1 0
Mean = 2 0
Variance = 3 0
4 0
3 Throw a coin 100 times, record the results. 5 0
Head = 1 6 1Tail = 0 7 1
8 1
9 0
10 0 0.3 0.21 0.233333333
Some useful equations: 11 0
Mean 12 0
13 1
14 0
Variance 15 1
16 0
17 1
Sample Mean 18 0
19 0
20 1 0.35 0.2275 0.239473684
21 0Sample Variance 22 1
23 0
24 1
25 0
Unbiased Variance 26 1
27 1
28 0
29 1
30 0 0.4 0.24 0.248275862
31 1
32 0
33 1
34 1
35 0
36 037 1
38 0
39 0
40 1 0.425 0.244375 0.250641026
41 1
42 0
43 1
44 1
45 0
46 1
47 1
48 0
49 1
50 1 0.48 0.2496 0.254693878
51 1
52 1
53 1
54 0
55 1
56 1
57 0
58 0
59 1
60 1 0.516666667 0.249722222 0.253954802
61 0
62 0
63 0
64 1
65 1
66 0
67 0
Unbiased Sample
VarianceSample Mean Sample Variance
Sample Mean Sample VarianceUnbiased Sample
Variance
Sample Mean Sample VarianceUnbiased Sample
Variance
Sample Mean Sample VarianceUnbiased Sample
Variance
Sample Mean Sample VarianceUnbiased Sample
Variance
Sample Mean Sample VarianceUnbiased Sample
Variance
1
1 nii
X Xn
2
2
1
1 nn ii
S X Xn
2
2
1 1
1
1
n
n iiS X X
n
[ ]E X p
( ) (1 )Var X p p
8/11/2019 Experiments Another Group
2/24
68 0
69 0
70 1 0.485714286 0.249795918 0.253416149
71 1
72 1
73 0
74 0
75 1
76 177 0
78 0
79 1
80 0 0.4875 0.24984375 0.253006329
81 1
82 0
83 0
84 0
85 1
86 0
87 1
88 1
89 1
90 1 0.5 0.25 0.25280898991 0
92 1
93 1
94 1
95 1
96 1
97 0
98 1
99 0
100 0 0.51 0.2499 0.252424242
The following data comes from another group 101 1
102 1
103 1
104 0105 1
106 0
107 1
108 0
109 0
110 1 0.518181818 0.249669421 0.251959967
111 0
112 1
113 1
114 0
115 0
116 1
117 0
118 1
119 0
120 0 0.508333333 0.249930556 0.252030812
121 0
122 0
123 1
124 0
125 0
126 1
127 0
128 1
129 1
130 1 0.507692308 0.249940828 0.251878354
131 1
132 0
133 1
134 0
135 0
Sample Mean Sample VarianceUnbiased Sampl
Variance
Sample Mean Sample VarianceUnbiased Sampl
Variance
Sample Mean Sample VarianceUnbiased Sampl
Variance
Sample Mean Sample VarianceUnbiased Sampl
Variance
Sample Mean Sample VarianceUnbiased Sampl
Variance
Sample Mean Sample Variance
Unbiased Sampl
Variance
Sample Mean Sample VarianceUnbiased Sampl
Variance
8/11/2019 Experiments Another Group
3/24
8/11/2019 Experiments Another Group
4/24
Sample Size 10 20 30 40 50 60 70 80 90 100
Sample Mean 0.3 0.35 0.4 0.425 0.48 0.517 0.486 0.488 0.5 0.51
Sample Var 0.21 0.228 0.24 0.244 0.25 0.25 0.25 0.25 0.25 0.25
Unbiased Var 0.233 0.239 0.248 0.251 0.255 0.254 0.253 0.253 0.253 0.252
0
0.1
0.2
0.3
0.4
0.5
0.6
0 10 20 30 40 50 60 70 80 90 100
Sample Mean Sample Var Unbiased Var
8/11/2019 Experiments Another Group
5/24
Binomial Distribution - Bin(5, 0.5)
1 Flip 5 fair coins at the same time and count Trial Outcome
the total number of head. 1 1
2 Compute the (population) mean and variance 2 4
Mean = 3 4
Variance = 4 2
5 4
3 Repeat 100 times, record the results. 6 2Head = 1 7 1
Tail = 0 8 1
9 1
10 3 2.3 1.61 1.788888889
Some useful equations: 11 3
Mean 12 1
13 3
14 4
Variance 15 4
16 1
17 3
Sample Mean 18 1
19 3
20 2 2.4 1.44 1.515789474
21 4Sample Variance 22 4
23 2
24 1
25 2
Unbiased Variance 26 3
27 2
28 2
29 0
30 1 2.3 1.476666667 1.527586207
31 3
32 2
33 5
34 3
35 3
36 137 2
38 3
39 3
40 2 2.4 1.39 1.425641026
41 2
42 3
43 1
44 3
45 4
46 2
47 2
48 3
49 3
50 2 2.42 1.2436 1.268979592
51 3
52 2
53 4
54 3
55 3
56 3
57 3
58 3
59 1
60 3 2.483333333 1.149722222 1.16920904
61 2
62 2
63 0
64 2
65 2
66 3
67 3
Sample Mean Sample VarianceUnbiased Sample
Variance
Sample Mean Sample VarianceUnbiased Sample
Variance
Sample Mean Sample VarianceUnbiased Sample
Variance
Sample Mean Sample VarianceUnbiased Sample
Variance
Sample Mean Sample VarianceUnbiased Sample
Variance
Sample Mean Sample VarianceUnbiased Sample
Variance
1
1 nii
X Xn
2
2
1
1 nn ii
S X Xn
2
2
1 1
1
1
n
n iiS X X
n
[ ]E X np
( ) (1 )Var X np p
8/11/2019 Experiments Another Group
6/24
68 2
69 3
70 4 2.457142857 1.133877551 1.150310559
71 2
72 2
73 2
74 2
75 3
76 377 3
78 3
79 3
80 3 2.475 1.024375 1.037341772
81 0
82 0
83 3
84 3
85 3
86 4
87 4
88 2
89 3
90 2 2.466666667 1.115555556 1.12808988891 2
92 2
93 2
94 1
95 4
96 4
97 4
98 1
99 0
100 2 2.44 1.1864 1.198383838
The following data comes from another group 101
102
103
104105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
Sample Mean Sample VarianceUnbiased Sampl
Variance
Sample Mean Sample VarianceUnbiased Sampl
Variance
Sample Mean Sample VarianceUnbiased Sampl
Variance
Sample Mean Sample VarianceUnbiased Sampl
Variance
Sample Mean Sample VarianceUnbiased Sampl
Variance
Sample Mean Sample Variance
Unbiased Sampl
Variance
Sample Mean Sample VarianceUnbiased Sampl
Variance
8/11/2019 Experiments Another Group
7/24
139
140
141
142
143
144
145
146
147148
149
150
151
152
153
154
155
156
157
158
159
160
161162
163
164
165
166
167
168
169
170
171
172
173
174
175176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
Variance
Sample Mean Sample VarianceUnbiased Sampl
Variance
Sample Mean Sample VarianceUnbiased Sampl
Variance
Sample Mean Sample VarianceUnbiased Sampl
Variance
Sample Mean Sample VarianceUnbiased Sampl
Variance
Sample Mean Sample VarianceUnbiased Sampl
Variance
Sample Mean Sample VarianceUnbiased Sampl
Variance
8/11/2019 Experiments Another Group
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Sample Size 10 20 30 40 50 60 70 80 90 100
Sample Mean 2.3 2.4 2.3 2.4 2.42 2.483 2.457 2.475 2.467 2.44
Sample Var 1.61 1.44 1.477 1.39 1.244 1.15 1.134 1.024 1.116 1.186
Unbiased Var 1.789 1.516 1.528 1.426 1.269 1.169 1.15 1.037 1.128 1.198
0
0.5
1
1.5
2
2.5
3
0 10 20 30 40 50 60 70 80 90 100
Sample Mean Sample Var Unbiased Var
8/11/2019 Experiments Another Group
9/24
1 ~ 100 1 2 3 4 5
Count 14 30 35 15 1
1 ~ 200 1 2 3 4 5
Count 14 30 35 15 1
14
30
35
15
1
0
5
10
15
20
25
30
35
40
1 2 3 4 5
14
30
35
15
1
0
5
10
15
20
25
30
35
40
1 2 3 4 5
8/11/2019 Experiments Another Group
10/24
Binomial Distribution - Bin(10, 0.5)
1 Flip 10 fair coins at the same time and count Trial Outcome
the total number of head. 1 1
2 Compute the (population) mean and variance 2 5
Mean = 3 6
Variance = 4 5
5 4
3 Repeat 100 times, record the results. 6 2Head = 1 7 4
Tail = 0 8 6
9 5
10 7 4.5 3.05 3.388888889
Some useful equations: 11 8
Mean 12 8
13 6
14 4
Variance 15 6
16 6
17 4
Sample Mean 18 3
19 3
20 7 5 3.4 3.578947368
21 7Sample Variance 22 4
23 5
24 3
25 2
Unbiased Variance 26 4
27 5
28 3
29 4
30 1 4.6 3.44 3.55862069
31 4
32 3
33 4
34 4
35 3
36 2
37 3
38 3
39 7
40 5 4.4 3.14 3.220512821
41 5
42 5
43 3
44 2
45 7
46 7
47 8
48 9
49 4
50 5 4.62 3.5956 3.668979592
51 5
52 6
53 8
54 7
55 4
56 5
57 4
58 2
59 5
60 4 4.683333333 3.449722222 3.50819209
61 5
62 3
63 6
64 5
65 4
66 4
67 3
Sample Mean Sample VarianceUnbiased Sample
Variance
Sample Mean Sample VarianceUnbiased Sample
Variance
Sample Mean Sample VarianceUnbiased Sample
Variance
Sample Mean Sample VarianceUnbiased Sample
Variance
Sample Mean Sample VarianceUnbiased Sample
Variance
Sample Mean Sample VarianceUnbiased Sample
Variance
1
1 nii
X Xn
2
2
1
1 nn ii
S X Xn
2
2
1 1
1
1
n
n iiS X X
n
[ ]E X np
( ) (1 )Var X np p
8/11/2019 Experiments Another Group
11/24
68 5
69 5
70 7 4.685714286 3.158367347 3.204140787
71 4
72 5
73 4
74 6
75 4
76 577 5
78 4
79 6
80 10 4.7625 3.18109375 3.221360759
81 7
82 8
83 5
84 6
85 4
86 5
87 4
88 6
89 4
90 8 4.866666667 3.16 3.19550561891 7
92 8
93 5
94 3
95 5
96 6
97 6
98 4
99 3
100 4 4.89 3.0979 3.129191919
The following data comes from another group 101 2
102 5
103 3
104 2105 7
106 5
107 8
108 2
109 4
110 6 4.845454545 3.221570248 3.251125938
111 7
112 3
113 2
114 6
115 7
116 4
117 8
118 9
119 1
120 2 4.85 3.560833333 3.590756303
121 5
122 3
123 6
124 7
125 6
126 5
127 4
128 3
129 6
130 7 4.876923077 3.446390533 3.473106738
131 4
132 3
133 5
134 6
135 4
Sample Mean Sample VarianceUnbiased Sampl
Variance
Sample Mean Sample VarianceUnbiased Sampl
Variance
Sample Mean Sample VarianceUnbiased Sampl
Variance
Sample Mean Sample VarianceUnbiased Sampl
Variance
Sample Mean Sample VarianceUnbiased Sampl
Variance
Sample Mean Sample Variance
Unbiased Sampl
Variance
Sample Mean Sample VarianceUnbiased Sampl
Variance
8/11/2019 Experiments Another Group
12/24
139 4
140 7 4.864285714 3.317295918 3.341161357
141 2
142 4
143 7
144 8
145 5
146 3
147 8148 4
149 7
150 3 4.88 3.398933333 3.421744966
151 4
152 7
153 9
154 3
155 6
156 4
157 5
158 7
159 5
160 3 4.90625 3.409960938 3.431407233
161 4162 5
163 7
164 6
165 5
166 2
167 3
168 7
169 8
170 5 4.923529412 3.400034602 3.42015315
171 5
172 4
173 3
174 6
175 7176 6
177 5
178 6
179 4
180 2 4.916666667 3.331944444 3.350558659
181 6
182 8
183 6
184 7
185 4
186 4
187 3
188 5
189 6
190 3 4.931578947 3.29531856 3.312754107
191 6
192 8
193 9
194 4
195 5
196 2
197 5
198 2
199 6
200 7 4.955 3.382975 3.399974874
Variance
Sample Mean Sample VarianceUnbiased Sampl
Variance
Sample Mean Sample VarianceUnbiased Sampl
Variance
Sample Mean Sample VarianceUnbiased Sampl
Variance
Sample Mean Sample VarianceUnbiased Sampl
Variance
Sample Mean Sample VarianceUnbiased Sampl
Variance
Sample Mean Sample VarianceUnbiased Sampl
Variance
8/11/2019 Experiments Another Group
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Sample Size 10 20 30 40 50 60 70 80 90 100
Sample Mean 4.5 5 4.6 4.4 4.62 4.683 4.686 4.763 4.867 4.89
Sample Var 3.05 3.4 3.44 3.14 3.596 3.45 3.158 3.181 3.16 3.098
Unbiased Var 3.389 3.579 3.559 3.221 3.669 3.508 3.204 3.221 3.196 3.129
0
1
2
3
4
5
6
0 10 20 30 40 50 60 70 80 90 100
Sample Mean Sample Var Unbiased Var
8/11/2019 Experiments Another Group
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1 ~ 100 1 2 3 4 5 6 7 8 9 10
Count 2 5 13 25 23 13 10 7 1 1
1 ~ 200 1 2 3 4 5 6 7 8 9 10
Count 3 15 27 41 40 30 25 14 4 1
2
5
13
25
23
13
10
7
1 1
0
5
10
15
20
25
30
1 2 3 4 5 6 7 8 9 10
14
30
35
15
1
0
5
10
15
20
25
30
35
40
1 2 3 4 5
3
15
27
4140
30
25
14
4
1
0
5
10
15
20
25
30
35
40
45
1 2 3 4 5 6 7 8 9 10
8/11/2019 Experiments Another Group
15/24
Geometric Distribution - Geom(0.25)
1 Flip 3 coins at the same time. Record the total Trial Outcome
number of trials until you get all heads or tails. 1 3
2 Compute the (population) mean and variance 2 6
Mean = 3 4
Variance = 4 7
5 2
3 Repeat 100 times, record the results. 6 4
Head = 1 7 3
Tail = 0 8 8
(Record 15 if the number > 15.) 9 3
10 1 4.1 4.49 4.988888889
Some useful equations: 11 1
Mean 12 9
13 10
14 4
Variance 15 2
16 11
17 7
Sample Mean 18 1
19 1
20 5 4.6 9.44 9.936842105
21 6Sample Variance 22 9
23 3
24 2
25 1
Unbiased Variance 26 8
27 3
28 1
29 2
30 5 4.4 8.84 9.144827586
31 2
32 5
33 5
34 3
35 4
36 137 1
38 4
39 7
40 4 4.2 7.56 7.753846154
41 2
42 10
43 1
44 2
45 4
46 8
47 3
48 2
49 7
50 1 4.16 7.8944 8.055510204
51 752 3
53 2
54 1
55 2
56 1
57 3
58 2
59 9
60 13 4.183333333 9.016388889 9.16920904
61 12
62 7
63 4
64 4
65 5
66 267 5
Sample Mean Sample VarianceUnbiased Sample
Variance
Sample Mean Sample VarianceUnbiased Sample
Variance
Sample Mean Sample VarianceUnbiased Sample
Variance
Sample Mean Sample VarianceUnbiased Sample
Variance
Sample Mean Sample VarianceUnbiased Sample
Variance
Sample Mean Sample VarianceUnbiased Sample
Variance
1
1 nii
X X
n
2
2
1
1 nn ii
S X Xn
2
2
1 1
1
1
n
n iiS X X
n
[ ] 1/E X p
2( ) (1 ) /Var X p p
8/11/2019 Experiments Another Group
16/24
68 2
69 3
70 2 4.242857143 8.955306122 9.085093168
71 1
72 5
73 2
74 6
75 12
76 177 1
78 1
79 6
80 4 4.2 9.26 9.37721519
81 1
82 3
83 3
84 10
85 3
86 4
87 3
88 1
89 4
90 1 4.1 9.001111111 9.10224719191 9
92 2
93 3
94 4
95 1
96 3
97 2
98 4
99 2
100 4 4.03 8.5891 8.675858586
The following data comes from another group 101
102
103
104105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
Sample Mean Sample VarianceUnbiased Sampl
Variance
Sample Mean Sample VarianceUnbiased Sampl
Variance
Sample Mean Sample VarianceUnbiased Sampl
Variance
Sample Mean Sample VarianceUnbiased Sampl
Variance
Sample Mean Sample VarianceUnbiased Sampl
Variance
Sample Mean Sample Variance
Unbiased Sampl
Variance
Sample Mean Sample VarianceUnbiased Sampl
Variance
8/11/2019 Experiments Another Group
17/24
139
140
141
142
143
144
145
146
147148
149
150
151
152
153
154
155
156
157
158
159
160
161162
163
164
165
166
167
168
169
170
171
172
173
174
175176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
Variance
Sample Mean Sample VarianceUnbiased Sampl
Variance
Sample Mean Sample VarianceUnbiased Sampl
Variance
Sample Mean Sample VarianceUnbiased Sampl
Variance
Sample Mean Sample VarianceUnbiased Sampl
Variance
Sample Mean Sample VarianceUnbiased Sampl
Variance
Sample Mean Sample VarianceUnbiased Sampl
Variance
8/11/2019 Experiments Another Group
18/24
Sample Size 10 20 30 40 50 60 70 80 90 100
Sample Mean 4.1 4.6 4.4 4.2 4.16 4.183 4.243 4.2 4.1 4.03
Sample Var 4.49 9.44 8.84 7.56 7.894 9.016 8.955 9.26 9.001 8.589
Unbiased Var 4.989 9.937 9.145 7.754 8.056 9.169 9.085 9.377 9.102 8.676
0
2
4
6
8
10
12
0 10 20 30 40 50 60 70 80 90 100
Sample Mean Sample Var Unbiased Var
8/11/2019 Experiments Another Group
19/24
1 ~ 100 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Count 20 18 16 15 7 4 6 3 4 3 1 2 1 0 0
1 ~ 200 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Count 20 18 16 15 7 4 6 3 4 3 1 2 1 0 0
20
18
1615
7
4
6
34
3
12
10 0
0
5
10
15
20
25
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
14
30
35
15
1
0
5
10
15
20
25
30
35
40
1 2 3 4 5
20
18
1615
7
4
6
34
3
12
10 0
0
5
10
15
20
25
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
8/11/2019 Experiments Another Group
20/24
8/11/2019 Experiments Another Group
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8/11/2019 Experiments Another Group
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8/11/2019 Experiments Another Group
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Questions1 Describe the shapes of the graphs in each experiment. Explain for each graph, why the graphs have thes
2 Compare the biased and the unbiased versions of the sample variance. Which is closer to the true distri
3 Compare the two Binomial distributions, which result that you learned in the lectures explains the differ
4 In the Geometric distribution trials, on average we need to throw 4 times to get the result. If now I want
in order to optimize your chance to win, which number will you guess? Why? (10 marks)
5 Combine your experiment outcomes with another group's (i.e., assume another group's outcome is your
Compare the sample mean, sample variance (biased and unbiased) with the mean and variances of the d
6 In the turning pen experiment, do you see the law of large numbers and/or central limit theorem? How
Remark:
1 The completion of each experiment is 10 marks. So, a total of 40 marks is given for those experiments.
2 The turning pen experiment will be done later in the class.
3 Each student should submit an individual report on 10-Sep.
8/11/2019 Experiments Another Group
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shapes? (10 marks)
bution variance? (10 marks)
nces between them? How? (10 marks)
you to guess how many times it take to get all heads or all tails,
101~200).
istributions. What do you observe? Why? (10 marks)
nd why? (10 marks)
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