Week 5
Galih Kusuma Wijaya
2026-03-09
Distribution Estimation and Model Parameter
par(mfrow=c(3,1))
library(probs)##
## Attaching package: 'probs'## The following objects are masked from 'package:base':
##
## intersect, setdiff, unionset.seed(1312)
populasi = rgeom(20,0.1)
n1 = 2
ex_geol = urnsamples(populasi,
size=2,
replace=F,
ordered=F)
mean_geol = matrix(apply(ex_geol, 1, mean))
n2 = 5
ex_geol2 = urnsamples(populasi,
size=5,
replace=F,
ordered=F)
mean_geol2 = matrix(apply(ex_geol2, 1, mean))
n3 = 10
ex_geol3 = urnsamples(populasi,
size=10,
replace=F,
ordered=F)
mean_geol3 = matrix(apply(ex_geol3, 1, mean))
hist(mean_geol,main = paste("Hampiran Normal Terhadap Geomertrik (n = 2)", xlab='xbar'))
hist(mean_geol2,main = paste("Hampiran Normal Terhadap Geomertrik (n = 5)", xlab='xbar'))
hist(mean_geol3,main = paste("Hampiran Normal Terhadap Geomertrik (n = 10)", xlab='xbar'))
Normal approximation to the exponential distribution
set.seed(1)
populasi = rexp(20)
n1 = 2
ex_geol = urnsamples(populasi,
size=2,
replace=F,
ordered=F)
mean_geol = matrix(apply(ex_geol, 1, mean))
n2 = 5
ex_geol2 = urnsamples(populasi,
size=5,
replace=F,
ordered=F)
mean_geol2 = matrix(apply(ex_geol2, 1, mean))
n3 = 10
ex_geol3 = urnsamples(populasi,
size=10,
replace=F,
ordered=F)
mean_geol3 = matrix(apply(ex_geol3, 1, mean))
hist(mean_geol,main = paste("Hampiran Normal Terhadap Geomertrik (n = 2)", xlab='xbar'))
hist(mean_geol2,main = paste("Hampiran Normal Terhadap Geomertrik (n = 5)", xlab='xbar'))
hist(mean_geol3,main = paste("Hampiran Normal Terhadap Geomertrik (n = 10)", xlab='xbar'))
Normal approximation to the uniform distribution
set.seed(2)
populasi = runif(20)
n1 = 2
ex_geol = urnsamples(populasi,
size=2,
replace=F,
ordered=F)
mean_geol = matrix(apply(ex_geol, 1, mean))
n2 = 5
ex_geol2 = urnsamples(populasi,
size=5,
replace=F,
ordered=F)
mean_geol2 = matrix(apply(ex_geol2, 1, mean))
n3 = 10
ex_geol3 = urnsamples(populasi,
size=10,
replace=F,
ordered=F)
mean_geol3 = matrix(apply(ex_geol3, 1, mean))
hist(mean_geol,main = paste("Hampiran Normal Terhadap Geomertrik (n = 2)", xlab='xbar'))
hist(mean_geol2,main = paste("Hampiran Normal Terhadap Geomertrik (n = 5)", xlab='xbar'))
hist(mean_geol3,main = paste("Hampiran Normal Terhadap Geomertrik (n = 10)", xlab='xbar'))
Kesimpulan: Semakin besar ukuran contoh, maka sebaran rata-rata dari
contoh acak yang berasal dari sebaran geometrik, eksponensial, maupun
uniform akan mendekati sebaran normal. Hal ini ditunjukkan dari
histogram yang mana ketika n semakin besar akan semakin cenderung
membentuk kurva normal
Normal Distribution Spread
set.seed(1)
populasi = rnorm(20,5,sqrt(12))
n1 = 3
contoh_norm1 = urnsamples(populasi, size = 3, replace = F, ordered = F)
contoh_norm1## X1 X2 X3
## 1 2.829900 5.636159 2.105298
## 2 2.829900 5.636159 10.526215
## 3 2.829900 5.636159 6.141448
## 4 2.829900 5.636159 2.157814
## 5 2.829900 5.636159 6.688504
## 6 2.829900 5.636159 7.557632
## 7 2.829900 5.636159 6.994565
## 8 2.829900 5.636159 3.942104
## 9 2.829900 5.636159 10.236964
## 10 2.829900 5.636159 6.350457
## 11 2.829900 5.636159 2.847960
## 12 2.829900 5.636159 -2.671945
## 13 2.829900 5.636159 8.896875
## 14 2.829900 5.636159 4.844345
## 15 2.829900 5.636159 4.943915
## 16 2.829900 5.636159 8.269545
## 17 2.829900 5.636159 7.844794
## 18 2.829900 5.636159 7.057335
## 19 2.829900 2.105298 10.526215
## 20 2.829900 2.105298 6.141448
## 21 2.829900 2.105298 2.157814
## 22 2.829900 2.105298 6.688504
## 23 2.829900 2.105298 7.557632
## 24 2.829900 2.105298 6.994565
## 25 2.829900 2.105298 3.942104
## 26 2.829900 2.105298 10.236964
## 27 2.829900 2.105298 6.350457
## 28 2.829900 2.105298 2.847960
## 29 2.829900 2.105298 -2.671945
## 30 2.829900 2.105298 8.896875
## 31 2.829900 2.105298 4.844345
## 32 2.829900 2.105298 4.943915
## 33 2.829900 2.105298 8.269545
## 34 2.829900 2.105298 7.844794
## 35 2.829900 2.105298 7.057335
## 36 2.829900 10.526215 6.141448
## 37 2.829900 10.526215 2.157814
## 38 2.829900 10.526215 6.688504
## 39 2.829900 10.526215 7.557632
## 40 2.829900 10.526215 6.994565
## 41 2.829900 10.526215 3.942104
## 42 2.829900 10.526215 10.236964
## 43 2.829900 10.526215 6.350457
## 44 2.829900 10.526215 2.847960
## 45 2.829900 10.526215 -2.671945
## 46 2.829900 10.526215 8.896875
## 47 2.829900 10.526215 4.844345
## 48 2.829900 10.526215 4.943915
## 49 2.829900 10.526215 8.269545
## 50 2.829900 10.526215 7.844794
## 51 2.829900 10.526215 7.057335
## 52 2.829900 6.141448 2.157814
## 53 2.829900 6.141448 6.688504
## 54 2.829900 6.141448 7.557632
## 55 2.829900 6.141448 6.994565
## 56 2.829900 6.141448 3.942104
## 57 2.829900 6.141448 10.236964
## 58 2.829900 6.141448 6.350457
## 59 2.829900 6.141448 2.847960
## 60 2.829900 6.141448 -2.671945
## 61 2.829900 6.141448 8.896875
## 62 2.829900 6.141448 4.844345
## 63 2.829900 6.141448 4.943915
## 64 2.829900 6.141448 8.269545
## 65 2.829900 6.141448 7.844794
## 66 2.829900 6.141448 7.057335
## 67 2.829900 2.157814 6.688504
## 68 2.829900 2.157814 7.557632
## 69 2.829900 2.157814 6.994565
## 70 2.829900 2.157814 3.942104
## 71 2.829900 2.157814 10.236964
## 72 2.829900 2.157814 6.350457
## 73 2.829900 2.157814 2.847960
## 74 2.829900 2.157814 -2.671945
## 75 2.829900 2.157814 8.896875
## 76 2.829900 2.157814 4.844345
## 77 2.829900 2.157814 4.943915
## 78 2.829900 2.157814 8.269545
## 79 2.829900 2.157814 7.844794
## 80 2.829900 2.157814 7.057335
## 81 2.829900 6.688504 7.557632
## 82 2.829900 6.688504 6.994565
## 83 2.829900 6.688504 3.942104
## 84 2.829900 6.688504 10.236964
## 85 2.829900 6.688504 6.350457
## 86 2.829900 6.688504 2.847960
## 87 2.829900 6.688504 -2.671945
## 88 2.829900 6.688504 8.896875
## 89 2.829900 6.688504 4.844345
## 90 2.829900 6.688504 4.943915
## 91 2.829900 6.688504 8.269545
## 92 2.829900 6.688504 7.844794
## 93 2.829900 6.688504 7.057335
## 94 2.829900 7.557632 6.994565
## 95 2.829900 7.557632 3.942104
## 96 2.829900 7.557632 10.236964
## 97 2.829900 7.557632 6.350457
## 98 2.829900 7.557632 2.847960
## 99 2.829900 7.557632 -2.671945
## 100 2.829900 7.557632 8.896875
## 101 2.829900 7.557632 4.844345
## 102 2.829900 7.557632 4.943915
## 103 2.829900 7.557632 8.269545
## 104 2.829900 7.557632 7.844794
## 105 2.829900 7.557632 7.057335
## 106 2.829900 6.994565 3.942104
## 107 2.829900 6.994565 10.236964
## 108 2.829900 6.994565 6.350457
## 109 2.829900 6.994565 2.847960
## 110 2.829900 6.994565 -2.671945
## 111 2.829900 6.994565 8.896875
## 112 2.829900 6.994565 4.844345
## 113 2.829900 6.994565 4.943915
## 114 2.829900 6.994565 8.269545
## 115 2.829900 6.994565 7.844794
## 116 2.829900 6.994565 7.057335
## 117 2.829900 3.942104 10.236964
## 118 2.829900 3.942104 6.350457
## 119 2.829900 3.942104 2.847960
## 120 2.829900 3.942104 -2.671945
## 121 2.829900 3.942104 8.896875
## 122 2.829900 3.942104 4.844345
## 123 2.829900 3.942104 4.943915
## 124 2.829900 3.942104 8.269545
## 125 2.829900 3.942104 7.844794
## 126 2.829900 3.942104 7.057335
## 127 2.829900 10.236964 6.350457
## 128 2.829900 10.236964 2.847960
## 129 2.829900 10.236964 -2.671945
## 130 2.829900 10.236964 8.896875
## 131 2.829900 10.236964 4.844345
## 132 2.829900 10.236964 4.943915
## 133 2.829900 10.236964 8.269545
## 134 2.829900 10.236964 7.844794
## 135 2.829900 10.236964 7.057335
## 136 2.829900 6.350457 2.847960
## 137 2.829900 6.350457 -2.671945
## 138 2.829900 6.350457 8.896875
## 139 2.829900 6.350457 4.844345
## 140 2.829900 6.350457 4.943915
## 141 2.829900 6.350457 8.269545
## 142 2.829900 6.350457 7.844794
## 143 2.829900 6.350457 7.057335
## 144 2.829900 2.847960 -2.671945
## 145 2.829900 2.847960 8.896875
## 146 2.829900 2.847960 4.844345
## 147 2.829900 2.847960 4.943915
## 148 2.829900 2.847960 8.269545
## 149 2.829900 2.847960 7.844794
## 150 2.829900 2.847960 7.057335
## 151 2.829900 -2.671945 8.896875
## 152 2.829900 -2.671945 4.844345
## 153 2.829900 -2.671945 4.943915
## 154 2.829900 -2.671945 8.269545
## 155 2.829900 -2.671945 7.844794
## 156 2.829900 -2.671945 7.057335
## 157 2.829900 8.896875 4.844345
## 158 2.829900 8.896875 4.943915
## 159 2.829900 8.896875 8.269545
## 160 2.829900 8.896875 7.844794
## 161 2.829900 8.896875 7.057335
## 162 2.829900 4.844345 4.943915
## 163 2.829900 4.844345 8.269545
## 164 2.829900 4.844345 7.844794
## 165 2.829900 4.844345 7.057335
## 166 2.829900 4.943915 8.269545
## 167 2.829900 4.943915 7.844794
## 168 2.829900 4.943915 7.057335
## 169 2.829900 8.269545 7.844794
## 170 2.829900 8.269545 7.057335
## 171 2.829900 7.844794 7.057335
## 172 5.636159 2.105298 10.526215
## 173 5.636159 2.105298 6.141448
## 174 5.636159 2.105298 2.157814
## 175 5.636159 2.105298 6.688504
## 176 5.636159 2.105298 7.557632
## 177 5.636159 2.105298 6.994565
## 178 5.636159 2.105298 3.942104
## 179 5.636159 2.105298 10.236964
## 180 5.636159 2.105298 6.350457
## 181 5.636159 2.105298 2.847960
## 182 5.636159 2.105298 -2.671945
## 183 5.636159 2.105298 8.896875
## 184 5.636159 2.105298 4.844345
## 185 5.636159 2.105298 4.943915
## 186 5.636159 2.105298 8.269545
## 187 5.636159 2.105298 7.844794
## 188 5.636159 2.105298 7.057335
## 189 5.636159 10.526215 6.141448
## 190 5.636159 10.526215 2.157814
## 191 5.636159 10.526215 6.688504
## 192 5.636159 10.526215 7.557632
## 193 5.636159 10.526215 6.994565
## 194 5.636159 10.526215 3.942104
## 195 5.636159 10.526215 10.236964
## 196 5.636159 10.526215 6.350457
## 197 5.636159 10.526215 2.847960
## 198 5.636159 10.526215 -2.671945
## 199 5.636159 10.526215 8.896875
## 200 5.636159 10.526215 4.844345
## 201 5.636159 10.526215 4.943915
## 202 5.636159 10.526215 8.269545
## 203 5.636159 10.526215 7.844794
## 204 5.636159 10.526215 7.057335
## 205 5.636159 6.141448 2.157814
## 206 5.636159 6.141448 6.688504
## 207 5.636159 6.141448 7.557632
## 208 5.636159 6.141448 6.994565
## 209 5.636159 6.141448 3.942104
## 210 5.636159 6.141448 10.236964
## 211 5.636159 6.141448 6.350457
## 212 5.636159 6.141448 2.847960
## 213 5.636159 6.141448 -2.671945
## 214 5.636159 6.141448 8.896875
## 215 5.636159 6.141448 4.844345
## 216 5.636159 6.141448 4.943915
## 217 5.636159 6.141448 8.269545
## 218 5.636159 6.141448 7.844794
## 219 5.636159 6.141448 7.057335
## 220 5.636159 2.157814 6.688504
## 221 5.636159 2.157814 7.557632
## 222 5.636159 2.157814 6.994565
## 223 5.636159 2.157814 3.942104
## 224 5.636159 2.157814 10.236964
## 225 5.636159 2.157814 6.350457
## 226 5.636159 2.157814 2.847960
## 227 5.636159 2.157814 -2.671945
## 228 5.636159 2.157814 8.896875
## 229 5.636159 2.157814 4.844345
## 230 5.636159 2.157814 4.943915
## 231 5.636159 2.157814 8.269545
## 232 5.636159 2.157814 7.844794
## 233 5.636159 2.157814 7.057335
## 234 5.636159 6.688504 7.557632
## 235 5.636159 6.688504 6.994565
## 236 5.636159 6.688504 3.942104
## 237 5.636159 6.688504 10.236964
## 238 5.636159 6.688504 6.350457
## 239 5.636159 6.688504 2.847960
## 240 5.636159 6.688504 -2.671945
## 241 5.636159 6.688504 8.896875
## 242 5.636159 6.688504 4.844345
## 243 5.636159 6.688504 4.943915
## 244 5.636159 6.688504 8.269545
## 245 5.636159 6.688504 7.844794
## 246 5.636159 6.688504 7.057335
## 247 5.636159 7.557632 6.994565
## 248 5.636159 7.557632 3.942104
## 249 5.636159 7.557632 10.236964
## 250 5.636159 7.557632 6.350457
## 251 5.636159 7.557632 2.847960
## 252 5.636159 7.557632 -2.671945
## 253 5.636159 7.557632 8.896875
## 254 5.636159 7.557632 4.844345
## 255 5.636159 7.557632 4.943915
## 256 5.636159 7.557632 8.269545
## 257 5.636159 7.557632 7.844794
## 258 5.636159 7.557632 7.057335
## 259 5.636159 6.994565 3.942104
## 260 5.636159 6.994565 10.236964
## 261 5.636159 6.994565 6.350457
## 262 5.636159 6.994565 2.847960
## 263 5.636159 6.994565 -2.671945
## 264 5.636159 6.994565 8.896875
## 265 5.636159 6.994565 4.844345
## 266 5.636159 6.994565 4.943915
## 267 5.636159 6.994565 8.269545
## 268 5.636159 6.994565 7.844794
## 269 5.636159 6.994565 7.057335
## 270 5.636159 3.942104 10.236964
## 271 5.636159 3.942104 6.350457
## 272 5.636159 3.942104 2.847960
## 273 5.636159 3.942104 -2.671945
## 274 5.636159 3.942104 8.896875
## 275 5.636159 3.942104 4.844345
## 276 5.636159 3.942104 4.943915
## 277 5.636159 3.942104 8.269545
## 278 5.636159 3.942104 7.844794
## 279 5.636159 3.942104 7.057335
## 280 5.636159 10.236964 6.350457
## 281 5.636159 10.236964 2.847960
## 282 5.636159 10.236964 -2.671945
## 283 5.636159 10.236964 8.896875
## 284 5.636159 10.236964 4.844345
## 285 5.636159 10.236964 4.943915
## 286 5.636159 10.236964 8.269545
## 287 5.636159 10.236964 7.844794
## 288 5.636159 10.236964 7.057335
## 289 5.636159 6.350457 2.847960
## 290 5.636159 6.350457 -2.671945
## 291 5.636159 6.350457 8.896875
## 292 5.636159 6.350457 4.844345
## 293 5.636159 6.350457 4.943915
## 294 5.636159 6.350457 8.269545
## 295 5.636159 6.350457 7.844794
## 296 5.636159 6.350457 7.057335
## 297 5.636159 2.847960 -2.671945
## 298 5.636159 2.847960 8.896875
## 299 5.636159 2.847960 4.844345
## 300 5.636159 2.847960 4.943915
## 301 5.636159 2.847960 8.269545
## 302 5.636159 2.847960 7.844794
## 303 5.636159 2.847960 7.057335
## 304 5.636159 -2.671945 8.896875
## 305 5.636159 -2.671945 4.844345
## 306 5.636159 -2.671945 4.943915
## 307 5.636159 -2.671945 8.269545
## 308 5.636159 -2.671945 7.844794
## 309 5.636159 -2.671945 7.057335
## 310 5.636159 8.896875 4.844345
## 311 5.636159 8.896875 4.943915
## 312 5.636159 8.896875 8.269545
## 313 5.636159 8.896875 7.844794
## 314 5.636159 8.896875 7.057335
## 315 5.636159 4.844345 4.943915
## 316 5.636159 4.844345 8.269545
## 317 5.636159 4.844345 7.844794
## 318 5.636159 4.844345 7.057335
## 319 5.636159 4.943915 8.269545
## 320 5.636159 4.943915 7.844794
## 321 5.636159 4.943915 7.057335
## 322 5.636159 8.269545 7.844794
## 323 5.636159 8.269545 7.057335
## 324 5.636159 7.844794 7.057335
## 325 2.105298 10.526215 6.141448
## 326 2.105298 10.526215 2.157814
## 327 2.105298 10.526215 6.688504
## 328 2.105298 10.526215 7.557632
## 329 2.105298 10.526215 6.994565
## 330 2.105298 10.526215 3.942104
## 331 2.105298 10.526215 10.236964
## 332 2.105298 10.526215 6.350457
## 333 2.105298 10.526215 2.847960
## 334 2.105298 10.526215 -2.671945
## 335 2.105298 10.526215 8.896875
## 336 2.105298 10.526215 4.844345
## 337 2.105298 10.526215 4.943915
## 338 2.105298 10.526215 8.269545
## 339 2.105298 10.526215 7.844794
## 340 2.105298 10.526215 7.057335
## 341 2.105298 6.141448 2.157814
## 342 2.105298 6.141448 6.688504
## 343 2.105298 6.141448 7.557632
## 344 2.105298 6.141448 6.994565
## 345 2.105298 6.141448 3.942104
## 346 2.105298 6.141448 10.236964
## 347 2.105298 6.141448 6.350457
## 348 2.105298 6.141448 2.847960
## 349 2.105298 6.141448 -2.671945
## 350 2.105298 6.141448 8.896875
## 351 2.105298 6.141448 4.844345
## 352 2.105298 6.141448 4.943915
## 353 2.105298 6.141448 8.269545
## 354 2.105298 6.141448 7.844794
## 355 2.105298 6.141448 7.057335
## 356 2.105298 2.157814 6.688504
## 357 2.105298 2.157814 7.557632
## 358 2.105298 2.157814 6.994565
## 359 2.105298 2.157814 3.942104
## 360 2.105298 2.157814 10.236964
## 361 2.105298 2.157814 6.350457
## 362 2.105298 2.157814 2.847960
## 363 2.105298 2.157814 -2.671945
## 364 2.105298 2.157814 8.896875
## 365 2.105298 2.157814 4.844345
## 366 2.105298 2.157814 4.943915
## 367 2.105298 2.157814 8.269545
## 368 2.105298 2.157814 7.844794
## 369 2.105298 2.157814 7.057335
## 370 2.105298 6.688504 7.557632
## 371 2.105298 6.688504 6.994565
## 372 2.105298 6.688504 3.942104
## 373 2.105298 6.688504 10.236964
## 374 2.105298 6.688504 6.350457
## 375 2.105298 6.688504 2.847960
## 376 2.105298 6.688504 -2.671945
## 377 2.105298 6.688504 8.896875
## 378 2.105298 6.688504 4.844345
## 379 2.105298 6.688504 4.943915
## 380 2.105298 6.688504 8.269545
## 381 2.105298 6.688504 7.844794
## 382 2.105298 6.688504 7.057335
## 383 2.105298 7.557632 6.994565
## 384 2.105298 7.557632 3.942104
## 385 2.105298 7.557632 10.236964
## 386 2.105298 7.557632 6.350457
## 387 2.105298 7.557632 2.847960
## 388 2.105298 7.557632 -2.671945
## 389 2.105298 7.557632 8.896875
## 390 2.105298 7.557632 4.844345
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## 829 6.688504 2.847960 4.844345
## 830 6.688504 2.847960 4.943915
## 831 6.688504 2.847960 8.269545
## 832 6.688504 2.847960 7.844794
## 833 6.688504 2.847960 7.057335
## 834 6.688504 -2.671945 8.896875
## 835 6.688504 -2.671945 4.844345
## 836 6.688504 -2.671945 4.943915
## 837 6.688504 -2.671945 8.269545
## 838 6.688504 -2.671945 7.844794
## 839 6.688504 -2.671945 7.057335
## 840 6.688504 8.896875 4.844345
## 841 6.688504 8.896875 4.943915
## 842 6.688504 8.896875 8.269545
## 843 6.688504 8.896875 7.844794
## 844 6.688504 8.896875 7.057335
## 845 6.688504 4.844345 4.943915
## 846 6.688504 4.844345 8.269545
## 847 6.688504 4.844345 7.844794
## 848 6.688504 4.844345 7.057335
## 849 6.688504 4.943915 8.269545
## 850 6.688504 4.943915 7.844794
## 851 6.688504 4.943915 7.057335
## 852 6.688504 8.269545 7.844794
## 853 6.688504 8.269545 7.057335
## 854 6.688504 7.844794 7.057335
## 855 7.557632 6.994565 3.942104
## 856 7.557632 6.994565 10.236964
## 857 7.557632 6.994565 6.350457
## 858 7.557632 6.994565 2.847960
## 859 7.557632 6.994565 -2.671945
## 860 7.557632 6.994565 8.896875
## 861 7.557632 6.994565 4.844345
## 862 7.557632 6.994565 4.943915
## 863 7.557632 6.994565 8.269545
## 864 7.557632 6.994565 7.844794
## 865 7.557632 6.994565 7.057335
## 866 7.557632 3.942104 10.236964
## 867 7.557632 3.942104 6.350457
## 868 7.557632 3.942104 2.847960
## 869 7.557632 3.942104 -2.671945
## 870 7.557632 3.942104 8.896875
## 871 7.557632 3.942104 4.844345
## 872 7.557632 3.942104 4.943915
## 873 7.557632 3.942104 8.269545
## 874 7.557632 3.942104 7.844794
## 875 7.557632 3.942104 7.057335
## 876 7.557632 10.236964 6.350457
## 877 7.557632 10.236964 2.847960
## 878 7.557632 10.236964 -2.671945
## 879 7.557632 10.236964 8.896875
## 880 7.557632 10.236964 4.844345
## 881 7.557632 10.236964 4.943915
## 882 7.557632 10.236964 8.269545
## 883 7.557632 10.236964 7.844794
## 884 7.557632 10.236964 7.057335
## 885 7.557632 6.350457 2.847960
## 886 7.557632 6.350457 -2.671945
## 887 7.557632 6.350457 8.896875
## 888 7.557632 6.350457 4.844345
## 889 7.557632 6.350457 4.943915
## 890 7.557632 6.350457 8.269545
## 891 7.557632 6.350457 7.844794
## 892 7.557632 6.350457 7.057335
## 893 7.557632 2.847960 -2.671945
## 894 7.557632 2.847960 8.896875
## 895 7.557632 2.847960 4.844345
## 896 7.557632 2.847960 4.943915
## 897 7.557632 2.847960 8.269545
## 898 7.557632 2.847960 7.844794
## 899 7.557632 2.847960 7.057335
## 900 7.557632 -2.671945 8.896875
## 901 7.557632 -2.671945 4.844345
## 902 7.557632 -2.671945 4.943915
## 903 7.557632 -2.671945 8.269545
## 904 7.557632 -2.671945 7.844794
## 905 7.557632 -2.671945 7.057335
## 906 7.557632 8.896875 4.844345
## 907 7.557632 8.896875 4.943915
## 908 7.557632 8.896875 8.269545
## 909 7.557632 8.896875 7.844794
## 910 7.557632 8.896875 7.057335
## 911 7.557632 4.844345 4.943915
## 912 7.557632 4.844345 8.269545
## 913 7.557632 4.844345 7.844794
## 914 7.557632 4.844345 7.057335
## 915 7.557632 4.943915 8.269545
## 916 7.557632 4.943915 7.844794
## 917 7.557632 4.943915 7.057335
## 918 7.557632 8.269545 7.844794
## 919 7.557632 8.269545 7.057335
## 920 7.557632 7.844794 7.057335
## 921 6.994565 3.942104 10.236964
## 922 6.994565 3.942104 6.350457
## 923 6.994565 3.942104 2.847960
## 924 6.994565 3.942104 -2.671945
## 925 6.994565 3.942104 8.896875
## 926 6.994565 3.942104 4.844345
## 927 6.994565 3.942104 4.943915
## 928 6.994565 3.942104 8.269545
## 929 6.994565 3.942104 7.844794
## 930 6.994565 3.942104 7.057335
## 931 6.994565 10.236964 6.350457
## 932 6.994565 10.236964 2.847960
## 933 6.994565 10.236964 -2.671945
## 934 6.994565 10.236964 8.896875
## 935 6.994565 10.236964 4.844345
## 936 6.994565 10.236964 4.943915
## 937 6.994565 10.236964 8.269545
## 938 6.994565 10.236964 7.844794
## 939 6.994565 10.236964 7.057335
## 940 6.994565 6.350457 2.847960
## 941 6.994565 6.350457 -2.671945
## 942 6.994565 6.350457 8.896875
## 943 6.994565 6.350457 4.844345
## 944 6.994565 6.350457 4.943915
## 945 6.994565 6.350457 8.269545
## 946 6.994565 6.350457 7.844794
## 947 6.994565 6.350457 7.057335
## 948 6.994565 2.847960 -2.671945
## 949 6.994565 2.847960 8.896875
## 950 6.994565 2.847960 4.844345
## 951 6.994565 2.847960 4.943915
## 952 6.994565 2.847960 8.269545
## 953 6.994565 2.847960 7.844794
## 954 6.994565 2.847960 7.057335
## 955 6.994565 -2.671945 8.896875
## 956 6.994565 -2.671945 4.844345
## 957 6.994565 -2.671945 4.943915
## 958 6.994565 -2.671945 8.269545
## 959 6.994565 -2.671945 7.844794
## 960 6.994565 -2.671945 7.057335
## 961 6.994565 8.896875 4.844345
## 962 6.994565 8.896875 4.943915
## 963 6.994565 8.896875 8.269545
## 964 6.994565 8.896875 7.844794
## 965 6.994565 8.896875 7.057335
## 966 6.994565 4.844345 4.943915
## 967 6.994565 4.844345 8.269545
## 968 6.994565 4.844345 7.844794
## 969 6.994565 4.844345 7.057335
## 970 6.994565 4.943915 8.269545
## 971 6.994565 4.943915 7.844794
## 972 6.994565 4.943915 7.057335
## 973 6.994565 8.269545 7.844794
## 974 6.994565 8.269545 7.057335
## 975 6.994565 7.844794 7.057335
## 976 3.942104 10.236964 6.350457
## 977 3.942104 10.236964 2.847960
## 978 3.942104 10.236964 -2.671945
## 979 3.942104 10.236964 8.896875
## 980 3.942104 10.236964 4.844345
## 981 3.942104 10.236964 4.943915
## 982 3.942104 10.236964 8.269545
## 983 3.942104 10.236964 7.844794
## 984 3.942104 10.236964 7.057335
## 985 3.942104 6.350457 2.847960
## 986 3.942104 6.350457 -2.671945
## 987 3.942104 6.350457 8.896875
## 988 3.942104 6.350457 4.844345
## 989 3.942104 6.350457 4.943915
## 990 3.942104 6.350457 8.269545
## 991 3.942104 6.350457 7.844794
## 992 3.942104 6.350457 7.057335
## 993 3.942104 2.847960 -2.671945
## 994 3.942104 2.847960 8.896875
## 995 3.942104 2.847960 4.844345
## 996 3.942104 2.847960 4.943915
## 997 3.942104 2.847960 8.269545
## 998 3.942104 2.847960 7.844794
## 999 3.942104 2.847960 7.057335
## 1000 3.942104 -2.671945 8.896875
## 1001 3.942104 -2.671945 4.844345
## 1002 3.942104 -2.671945 4.943915
## 1003 3.942104 -2.671945 8.269545
## 1004 3.942104 -2.671945 7.844794
## 1005 3.942104 -2.671945 7.057335
## 1006 3.942104 8.896875 4.844345
## 1007 3.942104 8.896875 4.943915
## 1008 3.942104 8.896875 8.269545
## 1009 3.942104 8.896875 7.844794
## 1010 3.942104 8.896875 7.057335
## 1011 3.942104 4.844345 4.943915
## 1012 3.942104 4.844345 8.269545
## 1013 3.942104 4.844345 7.844794
## 1014 3.942104 4.844345 7.057335
## 1015 3.942104 4.943915 8.269545
## 1016 3.942104 4.943915 7.844794
## 1017 3.942104 4.943915 7.057335
## 1018 3.942104 8.269545 7.844794
## 1019 3.942104 8.269545 7.057335
## 1020 3.942104 7.844794 7.057335
## 1021 10.236964 6.350457 2.847960
## 1022 10.236964 6.350457 -2.671945
## 1023 10.236964 6.350457 8.896875
## 1024 10.236964 6.350457 4.844345
## 1025 10.236964 6.350457 4.943915
## 1026 10.236964 6.350457 8.269545
## 1027 10.236964 6.350457 7.844794
## 1028 10.236964 6.350457 7.057335
## 1029 10.236964 2.847960 -2.671945
## 1030 10.236964 2.847960 8.896875
## 1031 10.236964 2.847960 4.844345
## 1032 10.236964 2.847960 4.943915
## 1033 10.236964 2.847960 8.269545
## 1034 10.236964 2.847960 7.844794
## 1035 10.236964 2.847960 7.057335
## 1036 10.236964 -2.671945 8.896875
## 1037 10.236964 -2.671945 4.844345
## 1038 10.236964 -2.671945 4.943915
## 1039 10.236964 -2.671945 8.269545
## 1040 10.236964 -2.671945 7.844794
## 1041 10.236964 -2.671945 7.057335
## 1042 10.236964 8.896875 4.844345
## 1043 10.236964 8.896875 4.943915
## 1044 10.236964 8.896875 8.269545
## 1045 10.236964 8.896875 7.844794
## 1046 10.236964 8.896875 7.057335
## 1047 10.236964 4.844345 4.943915
## 1048 10.236964 4.844345 8.269545
## 1049 10.236964 4.844345 7.844794
## 1050 10.236964 4.844345 7.057335
## 1051 10.236964 4.943915 8.269545
## 1052 10.236964 4.943915 7.844794
## 1053 10.236964 4.943915 7.057335
## 1054 10.236964 8.269545 7.844794
## 1055 10.236964 8.269545 7.057335
## 1056 10.236964 7.844794 7.057335
## 1057 6.350457 2.847960 -2.671945
## 1058 6.350457 2.847960 8.896875
## 1059 6.350457 2.847960 4.844345
## 1060 6.350457 2.847960 4.943915
## 1061 6.350457 2.847960 8.269545
## 1062 6.350457 2.847960 7.844794
## 1063 6.350457 2.847960 7.057335
## 1064 6.350457 -2.671945 8.896875
## 1065 6.350457 -2.671945 4.844345
## 1066 6.350457 -2.671945 4.943915
## 1067 6.350457 -2.671945 8.269545
## 1068 6.350457 -2.671945 7.844794
## 1069 6.350457 -2.671945 7.057335
## 1070 6.350457 8.896875 4.844345
## 1071 6.350457 8.896875 4.943915
## 1072 6.350457 8.896875 8.269545
## 1073 6.350457 8.896875 7.844794
## 1074 6.350457 8.896875 7.057335
## 1075 6.350457 4.844345 4.943915
## 1076 6.350457 4.844345 8.269545
## 1077 6.350457 4.844345 7.844794
## 1078 6.350457 4.844345 7.057335
## 1079 6.350457 4.943915 8.269545
## 1080 6.350457 4.943915 7.844794
## 1081 6.350457 4.943915 7.057335
## 1082 6.350457 8.269545 7.844794
## 1083 6.350457 8.269545 7.057335
## 1084 6.350457 7.844794 7.057335
## 1085 2.847960 -2.671945 8.896875
## 1086 2.847960 -2.671945 4.844345
## 1087 2.847960 -2.671945 4.943915
## 1088 2.847960 -2.671945 8.269545
## 1089 2.847960 -2.671945 7.844794
## 1090 2.847960 -2.671945 7.057335
## 1091 2.847960 8.896875 4.844345
## 1092 2.847960 8.896875 4.943915
## 1093 2.847960 8.896875 8.269545
## 1094 2.847960 8.896875 7.844794
## 1095 2.847960 8.896875 7.057335
## 1096 2.847960 4.844345 4.943915
## 1097 2.847960 4.844345 8.269545
## 1098 2.847960 4.844345 7.844794
## 1099 2.847960 4.844345 7.057335
## 1100 2.847960 4.943915 8.269545
## 1101 2.847960 4.943915 7.844794
## 1102 2.847960 4.943915 7.057335
## 1103 2.847960 8.269545 7.844794
## 1104 2.847960 8.269545 7.057335
## 1105 2.847960 7.844794 7.057335
## 1106 -2.671945 8.896875 4.844345
## 1107 -2.671945 8.896875 4.943915
## 1108 -2.671945 8.896875 8.269545
## 1109 -2.671945 8.896875 7.844794
## 1110 -2.671945 8.896875 7.057335
## 1111 -2.671945 4.844345 4.943915
## 1112 -2.671945 4.844345 8.269545
## 1113 -2.671945 4.844345 7.844794
## 1114 -2.671945 4.844345 7.057335
## 1115 -2.671945 4.943915 8.269545
## 1116 -2.671945 4.943915 7.844794
## 1117 -2.671945 4.943915 7.057335
## 1118 -2.671945 8.269545 7.844794
## 1119 -2.671945 8.269545 7.057335
## 1120 -2.671945 7.844794 7.057335
## 1121 8.896875 4.844345 4.943915
## 1122 8.896875 4.844345 8.269545
## 1123 8.896875 4.844345 7.844794
## 1124 8.896875 4.844345 7.057335
## 1125 8.896875 4.943915 8.269545
## 1126 8.896875 4.943915 7.844794
## 1127 8.896875 4.943915 7.057335
## 1128 8.896875 8.269545 7.844794
## 1129 8.896875 8.269545 7.057335
## 1130 8.896875 7.844794 7.057335
## 1131 4.844345 4.943915 8.269545
## 1132 4.844345 4.943915 7.844794
## 1133 4.844345 4.943915 7.057335
## 1134 4.844345 8.269545 7.844794
## 1135 4.844345 8.269545 7.057335
## 1136 4.844345 7.844794 7.057335
## 1137 4.943915 8.269545 7.844794
## 1138 4.943915 8.269545 7.057335
## 1139 4.943915 7.844794 7.057335
## 1140 8.269545 7.844794 7.057335mean_norm1 = matrix(apply(contoh_norm1, 1, mean))
mean_xbar1 = mean(mean_norm1)
var_xbar1 = var(mean_norm1)
n2 = 4
contoh_norm2 = urnsamples(populasi, size = 4, replace = F, ordered = F)
mean_norm2 = matrix(apply(contoh_norm2, 1, mean))
mean_xbar2 = mean(mean_norm2)
var_xbar2 = var(mean_norm2)
n3 = 15
contoh_norm3 = urnsamples(populasi, size = 15, replace = F, ordered = F)
mean_norm3 = matrix(apply(contoh_norm3, 1, mean))
mean_xbar3 = mean(mean_norm3)
var_xbar3 = var(mean_norm3)
hist(mean_norm1,main = paste("(n = 3)"),xlab = "xbar")
hist(mean_norm2,main = paste("(n = 4)"),xlab = "xbar")
hist(mean_norm3,main = paste("(n = 15)"),xlab = "xbar")
hasil = data.frame("."=c("mean","varian"),"Populasi"=c(5,12),"n=3"=c(mean_xbar1,var_xbar1),"n=4"=c(mean_xbar2,var_xbar2),"n=15"=c(mean_xbar3,var_xbar3))
hasil## . Populasi n.3 n.4 n.15
## 1 mean 5 5.659994 5.659994 5.6599941
## 2 varian 12 2.838200 2.002091 0.1668172Kesimpulan: Bisa kita lihat pada output di atas bahwa contoh acak yang diambil dari populasi jika ukuran contoh semakin besar, maka mean sampel akan semakin mendekati mean populasi dan ragamnya semakin mendekati σ2/n
Unbiasedness of an parameter estimator
Mean
set.seed(123)
n = 10
populasi1 = rnorm(20)
mean_pop1 = mean(populasi1)
sampel_normal1 = urnsamples(populasi1, size = 10, replace = F, ordered = F)
mean_normal1 = matrix(apply(sampel_normal1, 1, mean))
median_normal1 = matrix(apply(sampel_normal1, 1, median))
harapan_mean_norm1 = mean(mean_normal1)
harapan_median_norm1 = mean(median_normal1)
set.seed(23)
n = 10
populasi2 = rexp(20)
mean_pop2 = mean(populasi2)
sampel_exp1 = urnsamples(populasi2, size = 10, replace = F, ordered = F)
mean_exp1 = matrix(apply(sampel_exp1, 1, mean))
median_exp1 = matrix(apply(sampel_exp1, 1, median))
harapan_mean_exp1 = mean(mean_exp1)
harapan_median_exp1 = mean(median_exp1)
set.seed(3)
n = 10
populasi3 = runif(20)
mean_pop3 = mean(populasi3)
sampel_unif1 = urnsamples(populasi3, size = 10, replace = F, ordered = F)
mean_unif1 = matrix(apply(sampel_unif1, 1, mean))
median_unif1 = matrix(apply(sampel_unif1, 1, median))
harapan_mean_unif1 = mean(mean_unif1)
harapan_median_unif1 = mean(median_unif1)
hasil = data.frame("Hasil"=c("mean_populasi","harapan_mean_contoh","harapan_median_contoh"),"Sebaran Normal"=c(mean_pop1,harapan_mean_norm1,harapan_median_norm1),"Sebaran Eksponensial"=c(mean_pop2,harapan_mean_exp1,harapan_median_exp1),"Sebaran Seragam"=c(mean_pop3,harapan_mean_unif1,harapan_median_unif1))
hasil## Hasil Sebaran.Normal Sebaran.Eksponensial Sebaran.Seragam
## 1 mean_populasi 0.1416238 0.6906160 0.5160432
## 2 harapan_mean_contoh 0.1416238 0.6906160 0.5160432
## 3 harapan_median_contoh 0.1174878 0.6081886 0.5358320Ragam
set.seed(8)
n = 10
populasi = rnorm(20)
sigma2 = var(populasi)*(20-1)/20 #fungsi var pada R adalah varian contoh (penyebut n-1) sehingga perlu dikali (n-1)/n
sampel = urnsamples(populasi, size = 10, replace = F, ordered = F)
s2.n1 = matrix(apply(sampel, 1, var))
E.s2.n1 = mean(s2.n1)
s2.n = s2.n1*(10-1)/10
E.s2.n = mean(s2.n)
set.seed(88)
n = 10
populasi2 = rexp(20)
sigma2.exp = var(populasi2)*(20-1)/20
sampel_exp = urnsamples(populasi2, size = 10, replace = F, ordered = F)
s2.n1.exp = matrix(apply(sampel_exp, 1, var))
E.s2.n1.exp = mean(s2.n1.exp)
s2.n.exp = s2.n1.exp*(10-1)/10
E.s2.n.exp = mean(s2.n.exp)
hasil = data.frame( "." = c("ragam populasi","nilai harapan ragam contoh (n-1)","nilai harapan ragam contoh (n)"),
"Sebaran Normal" = c(sigma2, E.s2.n1, E.s2.n),"Sebaran Eksponensial" = c(sigma2.exp, E.s2.n1.exp, E.s2.n.exp))
hasil## . Sebaran.Normal Sebaran.Eksponensial
## 1 ragam populasi 0.9816185 1.035190
## 2 nilai harapan ragam contoh (n-1) 1.0332826 1.089673
## 3 nilai harapan ragam contoh (n) 0.9299544 0.980706Interval Kepercayaan
n1 = 10
k = 100
alpha = 0.05
mu = 50
std = 10
set.seed(123)
sampel.norm1 = matrix(rnorm(n1*k,mu,std),k)
xbar.norm1 = apply(sampel.norm1,1,mean)
s.norm1 = apply(sampel.norm1,1,sd)
SE.norm1 = s.norm1/sqrt(n1)
z.norm1 = qnorm(1-alpha/2)
SK.norm1 = (xbar.norm1-z.norm1*SE.norm1 < mu & mu < xbar.norm1+z.norm1*SE.norm1)
x.norm1 = sum(SK.norm1)/k
n2 = 30
k = 100 #ulangan
alpha = 0.05
mu = 50
std = 10
set.seed(123)
sampel.norm2 = matrix(rnorm(n2*k,mu,std),k)
xbar.norm2 = apply(sampel.norm2,1,mean)
s.norm2 = apply(sampel.norm2,1,sd)
SE.norm2 = s.norm2/sqrt(n2)
z.norm2 = qnorm(1-alpha/2)
SK.norm2 = (xbar.norm2-z.norm2*SE.norm2 < mu & mu < xbar.norm2+z.norm2*SE.norm2)
x.norm2 = sum(SK.norm2)/k
n3 = 100
k = 100 #ulangan
alpha = 0.05
mu = 50
std = 10
set.seed(123)
sampel.norm3 = matrix(rnorm(n3*k,mu,std),k)
xbar.norm3 = apply(sampel.norm3,1,mean)
s.norm3 = apply(sampel.norm3,1,sd)
SE.norm3 = s.norm3/sqrt(n3)
z.norm3 = qnorm(1-alpha/2)
SK.norm3 = (xbar.norm3-z.norm3*SE.norm3 < mu & mu < xbar.norm3+z.norm3*SE.norm3)
x.norm3 = sum(SK.norm3)/k
hasil = data.frame("n" =c(10,30,100),"Ketepatan SK Sebaran Normal"=c(x.norm1, x.norm2, x.norm3))
hasil## n Ketepatan.SK.Sebaran.Normal
## 1 10 0.93
## 2 30 0.93
## 3 100 0.96matplot(rbind (xbar.norm2-z.norm2*SE.norm2, xbar.norm2+z.norm2*SE.norm2), rbind(1:k,1:k), col=ifelse(SK.norm2,"blue","red"), type = "l", lty = 1,main='Selang Kepercayaan 95% (n=100)', xlab='SK', ylab='banyak ulangan')
abline(v=mu)
- Visualisasi di atas merupakan hasil dari simulasi dengan interval
kepercayaan 95% untuk rata-rata populasi = 50 dan ukuran sampel = 100 -
Simulasi dilakukan sebanyak 100 kali, sehingga terdapat 100 selang
kepercayaan yang dihasilkan.