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# Random Sampling - Random Samples

Feb 01, 2017

## Documents

lydung

• 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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