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Random Sampling - Random Samples
Feb 01, 2017
Random Sampling- Random Samples
Why do we need Random Samples?Many business applications
-We will have a random variable X such that the probability distribution & expected value is unknown-The only way to make use of probability is to estimate E(X) and if possible Fx or fx
-This can be done with random sampling
Random SamplesX come from some random process. x results from a trial of the process (observation of X ) a set {x1, x2, , xn} of n independent observations of the same random variable X is called a random sample of size n.
Sample MeanWhat does a random sample tell us about a random variable?Consider random sample
set {x1, x2, , xn}SAMPLE MEAN
Example
Shift observed
1
2
3
4
5
6
7
8
9
10
Number of stoppages
2
11
6
8
6
5
10
4
8
3
Important can be used as an estimate of the parameter E(X).
In general, the larger the sample size n, the better will be the estimate
APPROXIMATING MASS AND DENSITY FUNCTIONS
If we have a large enough sample, we can group the data and form a histogram that approximates the probability mass function (for a finite random variable)or the probability density function (for a continuous random variable).
We have used bins of width 1 and have plotted relative frequencies.
The relative frequency of each value of X in the sample gives an estimate for the probability that X will assume that value. Hence, the relative frequency of a value x in the sample approximates P(X = x) = fX(x) [p.m.f - Discrete random variables]APPROXIMATING MASS FUNCTIONS-Discrete random variables
APPROXIMATING DENSITY FUNCTION-Continuous random variablesRecall that the p.d.f can be used to find probabilities P(a X b) is equal to the area under the curve of the p.d.f over the interval [a,b]If we want to use a histogram approximate the p.d.f then
Relative frequency of a bin= Area of the corresponding rectangle
ImportantBut we know
Area of rectangle=width x height
But Relative frequency of a bin= Area of the corresponding rectangle
Now the area of each rectangle represents the probabilityNow we must plot the adjusted relative frequencies against the mid points of the bins
Approximating the p.d.f. (Disney)
Since the width is 0.030.0024/0.03
(0.73+0.76)/2Histogram function is used for normalized ratios( Rnorm)
Histogram
BinsFrequencyRelative FrequencyAdjusted Rel. Freq.(Height)Midpoint
0.7610.002400.079940.745
0.7900.000000.000000.775
0.8200.000000.000000.805
0.8510.002400.079940.835
0.8810.002400.079940.865
0.9120.004800.159870.895
0.94230.055161.838530.925
0.97560.134294.396480.955
11260.3021610.071940.985
1.031200.287779.672261.015
1.06530.127104.236611.045
1.09200.047961.598721.075
1.1290.021580.719421.105
1.1540.009590.319741.135
1.1810.002400.079941.165
More00.000000.00000
Sum:417133.333
Area0.31
Histogram
0.07993605120.0799360512
00
00
0.07993605120.0799360512
0.07993605120.0799360512
0.15987210230.1598721023
1.83852917671.8385291767
4.39648281374.3964828137
10.07194244610.071942446
9.67226219029.6722621902
4.23661071144.2366107114
1.59872102321.5987210232
0.71942446040.7194244604
0.31974420460.3197442046
0.07993605120.0799360512
&A
Page &P
Approximation of p.d.f.
Norm Ratios
Normalized Ratio
0.96231
1.08241
1.01193
0.98505
0.93713
1.10230
0.97364
1.01044
1.09439
1.00311
1.00743
1.01077
0.94176
1.02023
1.03002
1.04084
0.75672
0.97689
0.94697
0.95310
1.03786
0.93501
1.02895
0.98297
0.99999
0.95666
1.01620
0.95803
0.98860
0.98171
0.93137
1.00298
0.97069
1.00010
1.04141
1.01053
0.99823
0.98516
1.07691
1.01994
0.99503
1.02691
1.00978
0.92485
0.98987
0.98865
0.94545
1.00075
1.03790
1.03032
0.91092
1.03452
1.00086
1.12639
1.05562
0.89887
0.94837
1.03723
1.00728
0.98438
0.95154
0.83139
1.10816
0.94217
0.91397
0.97583
1.06260
1.02058
0.95422
0.97248
1.03093
1.02183
0.97332
0.96195
0.95617
1.11934
1.01056
0.99888
0.98407
0.97794
0.97837
0.96094
0.98985
1.02670
1.01298
0.96996
1.00500
1.03047
0.90575
1.03142
1.08569
0.94705
0.99281
0.99435
1.12451
1.04692
0.96120
1.17630
0.87211
0.93617
0.99232
1.03629
1.12101
0.97467
1.07719
1.06298
1.00317
1.02985
1.00612
0.99169
1.02382
1.02925
1.06621
1.03871
0.91633
1.01567
1.07919
0.95196
0.99848
0.96325
0.93029
1.01233
0.99664
0.95810
0.97377
1.09772
1.06506
0.92404
1.02685
0.95647
1.01245
0.97459
0.95256
0.97223
1.05515
0.94173
1.05038
0.98617
1.00313
0.97804
0.94376
0.91917
1.03259
0.96627
1.09411
0.95523
0.93827
0.97114
1.00586
1.01664
1.03001
0.96184
1.03355
1.03676
0.96591
0.94679
1.10444
1.08429
0.98047
0.99076
0.96177
1.01879
0.98319
1.11730
0.97741
0.96169
1.12185
0.96959
1.08712
1.07539
0.93892
0.98424
1.00132
0.98919
0.91259
0.92133
0.91993
1.04612
0.95019
0.96802
0.93436
0.93509
1.03166
1.07345
0.96211
1.02619
0.93370
1.00215
1.01157
0.97255
1.05029
0.92461
0.94965
1.02805
1.07605
1.03842
0.98341
1.03326
1.00121
1.00831
1.00243
0.94248
0.97756
1.02524
1.02536
1.01467
1.08253
1.02488
1.00018
0.96379
1.04687
0.99105
1.02224
0.99489
0.98900
1.00086
1.09188
1.00322
1.04667
0.99888
0.99585
0.98769
0.99073
1.06020
0.99340
1.03534
0.98529
1.01678
0.98207
1.01845
0.96651
0.98181
1.01865
1.04006
0.99152
0.99003
0.96525
0.97076
0.98391
1.02499
0.99277
0.98236
1.02824
0.98216
1.02220
1.03765
1.00702
1.05933
0.98199
0.98221
1.01239
0.98884
0.98243
1.02245
0.98395
0.96367
1.04054
1.02804
1.02168
1.01673
1.03019
0.98059
0.97081
0.98333
1.02912
0.98139
0.96673
1.02314
0.99715
1.05362
1.04468
0.98755
1.01811
1.01450
0.99865
1.01674
0.99689
1.03797
1.06467
0.99888
0.97532
1.00318
1.01638
0.98379
1.07047
0.94223
1.00767
0.95861
0.94321
0.99689
1.02327
0.97121
1.04003
0.97087
1.01929
1.03263
1.00311
0.95241
0.97907
1.02739
0.94474
1.01649
0.98346
0.93570
1.05411
0.99320
1.01426
1.01252
1.02079
1.01097
1.01945
1.01137
0.99888
0.97251
1.01120
1.00246
0.99274
1.00195
0.99377
0.97887
1.05373
0.99677
1.01819
1.02081
0.99451
0.99452
1.00487
0.99669
1.00546
0.99018
1.04911
0.97655
0.98566
0.95668
1.04755
0.97302
1.00977
1.04922
0.99660
0.98759
0.99826
0.93165
1.03818
0.99670
1.02572
1.00790
1.02434
0.99198
1.00816
0.97179
1.01029
1.00116
0.99432
1.02628
0.96284
1.00114
0.98551
1.04307
1.00121
1.00710
1.00717
1.01330
1.03370
1.06237
1.01228
1.01246
0.98547
1.00591
1.04394
1.03993
0.97876
1.01051
1.01363
1.00780
1.06555
0.98325
1.03447
0.99566
1.02198
0.97820
0.97989
0.95068
0.98112
1.02623
0.96369
1.00775
0.96736
1.02837
0.99888
1.01385
0.99888
0.98413
0.99770
1.01078
0.94005
1.01308
0.98487
1.00736
1.02496
1.03492
0.99888
0.97829
0.99594
1.04168
0.97968
0.99292
0.93221
0.95633
0.99888
1.01785
0.96232
1.05397
0.98797
1.01832
0.95378
0.99098
0.99625
Approximating the p.d.f. (Disney)
Normalized ratiosHeight
Chart1
0.07993605120.0799360512
00
00
0.07993605120.0799360512
0.07993605120.0799360512
0.15987210230.1598721023
1.83852917671.8385291767
4.47641886494.4764188649
10.07194244610.071942446
9.59232613919
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