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Random Sampling - Random Samples

Feb 01, 2017

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  • 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