Experiments Another Group

8/11/2019 Experiments Another Group

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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


Variance


Variance


Variance


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


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


Variance


Variance


Variance


Variance

Sample Mean Sample Variance

Unbiased Sampl

Variance


Variance


3/24


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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


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


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


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


23 2

24 1

25 2


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


Variance


Variance


Variance


Variance


Variance


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


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


Variance


Variance


Variance


Variance


Variance


Unbiased Sampl

Variance


Variance


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


Variance


Variance


Variance


Variance


Variance


Variance


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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



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


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


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


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


23 5

24 3

25 2


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


Variance


Variance


Variance


Variance


Variance


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


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


Variance


Variance


Variance


Variance


Variance


Unbiased Sampl

Variance


Variance


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


Variance


Variance


Variance


Variance


Variance


Variance


13/24

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



14/24

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


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


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


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


23 3

24 2

25 1


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


Variance


Variance


Variance


Variance


Variance


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


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


Variance


Variance


Variance


Variance


Variance


Unbiased Sampl

Variance


Variance


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


Variance


Variance


Variance


Variance


Variance


Variance


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



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


20/24


21/24


22/24


23/24

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.


24/24

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)

Documents

Experiments Another Group