~ubuntu-branches/ubuntu/trusty/gsl-ref-html/trusty

Committer: Bazaar Package Importer
Author(s): Dirk Eddelbuettel
Date: 2006-04-12 19:46:32 UTC
mfrom: (1.3.1 upstream) (3.1.1 dapper)
Revision ID: james.westby@ubuntu.com-20060412194632-c9lodpl075pv9si3

Tags: 1.8-1

* New upstream release 1.8
* As with previous releases, the sources were obtained from the FSF web
pages by means of a wget call (c.f. the debian/rules target 'upstream')

* debian/control: Standards-Version increased to 3.6.2
* debian/copyright: Updated FSF address
* debian/rules: Set DH_COMPAT=4

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2D-Histogram-Statistics.html

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gsl-ref_19.html

<HTML>

<HEAD>

<!-- This HTML file has been created by texi2html 1.54+ (gsl)

from ../gsl-ref.texi -->

<TITLE>GNU Scientific Library -- Reference Manual - Random Number Distributions</TITLE>

</HEAD>

<BODY>

Go to the <A HREF="gsl-ref_1.html">first</A>, <A HREF="gsl-ref_18.html">previous</A>, <A HREF="gsl-ref_20.html">next</A>, <A HREF="gsl-ref_50.html">last</A> section, <A HREF="gsl-ref_toc.html">table of contents</A>.

<HR>

<H1><A NAME="SEC286" HREF="gsl-ref_toc.html#TOC286">Random Number Distributions</A></H1>

This chapter describes functions for generating random variates and

computing their probability distributions. Samples from the

distributions described in this chapter can be obtained using any of the

random number generators in the library as an underlying source of

randomness. In the simplest cases a non-uniform distribution can be

obtained analytically from the uniform distribution of a random number

generator by applying an appropriate transformation. This method uses

one call to the random number generator.

More complicated distributions are created by the

acceptance-rejection method, which compares the desired

distribution against a distribution which is similar and known

analytically. This usually requires several samples from the generator.

The library also provides cumulative distribution functions and inverse

cumulative distribution functions, sometimes referred to as quantile

functions. The cumulative distribution functions and their inverses are

computed separately for the upper and lower tails of the distribution,

allowing full accuracy to be retained for small results.

The functions for random variates and probability density functions

described in this section are declared in <TT>'gsl_randist.h'</TT>. The

corresponding cumulative distribution functions are declared in

<TT>'gsl_cdf.h'</TT>.

<H2><A NAME="SEC287" HREF="gsl-ref_toc.html#TOC287">Introduction</A></H2>

Continuous random number distributions are defined by a probability

density function, p(x), such that the probability of x

occurring in the infinitesimal range x to x+dx is

p dx.

The cumulative distribution function for the lower tail is defined by,

P(x) = \int_{-\infty}^{x} dx' p(x')

</PRE>

and gives the probability of a variate taking a value less than x.

The cumulative distribution function for the upper tail is defined by,

Q(x) = \int_{x}^{+\infty} dx' p(x')

</PRE>

and gives the probability of a variate taking a value greater than x.

The upper and lower cumulative distribution functions are related by

P(x) + Q(x) = 1 and satisfy

0 <= P(x) <= 1,

0 <= Q(x) <= 1.

The inverse cumulative distributions,

x=P^{-1}(P) and

x=Q^{-1}(Q) give the values of x

which correspond to a specific value of P or {Q}.

100

They can be used to find confidence limits from probability values.

101

102

103

104

105

<H2><A NAME="SEC288" HREF="gsl-ref_toc.html#TOC288">The Gaussian Distribution</A></H2>

106

107

<DL>

108

<DT>Random: double gsl_ran_gaussian (const gsl_rng * <VAR>r</VAR>, double <VAR>sigma</VAR>)

109

110

111

This function returns a Gaussian random variate, with mean zero and

112

standard deviation <VAR>sigma</VAR>. The probability distribution for

113

Gaussian random variates is,

114

115

116

117

118

p(x) dx = {1 \over \sqrt{2 \pi \sigma^2}} \exp (-x^2 / 2\sigma^2) dx

119

</PRE>

120

121

122

for x in the range -\infty to +\infty. Use the

123

transformation z = \mu + x on the numbers returned by

124

<CODE>gsl_ran_gaussian</CODE> to obtain a Gaussian distribution with mean

125

\mu. This function uses the Box-Mueller algorithm which requires two

126

calls to the random number generator <VAR>r</VAR>.

127

</DL>

128

129

130

131

<DL>

132

<DT>Function: double gsl_ran_gaussian_pdf (double <VAR>x</VAR>, double <VAR>sigma</VAR>)

133

134

This function computes the probability density p(x) at <VAR>x</VAR>

135

for a Gaussian distribution with standard deviation <VAR>sigma</VAR>, using

136

the formula given above.

137

</DL>

138

139

140

141

142

143

<DL>

144

<DT>Function: double gsl_ran_gaussian_ratio_method (const gsl_rng * <VAR>r</VAR>, double <VAR>sigma</VAR>)

145

146

This function computes a Gaussian random variate using the

147

Kinderman-Monahan ratio method.

148

</DL>

149

150

151

152

<DL>

153

<DT>Random: double gsl_ran_ugaussian (const gsl_rng * <VAR>r</VAR>)

154

155

<DT>Function: double gsl_ran_ugaussian_pdf (double <VAR>x</VAR>)

156

157

<DT>Random: double gsl_ran_ugaussian_ratio_method (const gsl_rng * <VAR>r</VAR>)

158

159

These functions compute results for the unit Gaussian distribution. They

160

are equivalent to the functions above with a standard deviation of one,

161

<VAR>sigma</VAR> = 1.

162

</DL>

163

164

165

166

<DL>

167

<DT>Function: double gsl_cdf_gaussian_P (double <VAR>x</VAR>, double <VAR>sigma</VAR>)

168

169

<DT>Function: double gsl_cdf_gaussian_Q (double <VAR>x</VAR>, double <VAR>sigma</VAR>)

170

171

<DT>Function: double gsl_cdf_gaussian_Pinv (double <VAR>P</VAR>, double <VAR>sigma</VAR>)

172

173

<DT>Function: double gsl_cdf_gaussian_Qinv (double <VAR>Q</VAR>, double <VAR>sigma</VAR>)

174

175

These functions compute the cumulative distribution functions

176

P(x), Q(x) and their inverses for the Gaussian

177

distribution with standard deviation <VAR>sigma</VAR>.

178

</DL>

179

180

181

182

<DL>

183

<DT>Function: double gsl_cdf_ugaussian_P (double <VAR>x</VAR>)

184

185

<DT>Function: double gsl_cdf_ugaussian_Q (double <VAR>x</VAR>)

186

187

<DT>Function: double gsl_cdf_ugaussian_Pinv (double <VAR>P</VAR>)

188

189

<DT>Function: double gsl_cdf_ugaussian_Qinv (double <VAR>Q</VAR>)

190

191

These functions compute the cumulative distribution functions

192

P(x), Q(x) and their inverses for the unit Gaussian

193

distribution.

194

</DL>

195

196

197

198

199

<H2><A NAME="SEC289" HREF="gsl-ref_toc.html#TOC289">The Gaussian Tail Distribution</A></H2>

200

201

<DL>

202

<DT>Random: double gsl_ran_gaussian_tail (const gsl_rng * <VAR>r</VAR>, double <VAR>a</VAR>, double <VAR>sigma</VAR>)

203

204

205

This function provides random variates from the upper tail of a Gaussian

206

distribution with standard deviation <VAR>sigma</VAR>. The values returned

207

are larger than the lower limit <VAR>a</VAR>, which must be positive. The

208

method is based on Marsaglia's famous rectangle-wedge-tail algorithm (Ann

209

Math Stat 32, 894-899 (1961)), with this aspect explained in Knuth, v2,

210

3rd ed, p139,586 (exercise 11).

211

212

213

214

The probability distribution for Gaussian tail random variates is,

215

216

217

218

219

p(x) dx = {1 \over N(a;\sigma)} \exp (- x^2/(2 \sigma^2)) dx

220

</PRE>

221

222

223

for x > a where N(a;\sigma) is the normalization constant,

224

225

226

227

228

N(a;\sigma) = (1/2) erfc(a / sqrt(2 sigma^2)).

229

</PRE>

230

231

</DL>

232

233

234

<DL>

235

<DT>Function: double gsl_ran_gaussian_tail_pdf (double <VAR>x</VAR>, double <VAR>a</VAR>, double <VAR>sigma</VAR>)

236

237

This function computes the probability density p(x) at <VAR>x</VAR>

238

for a Gaussian tail distribution with standard deviation <VAR>sigma</VAR> and

239

lower limit <VAR>a</VAR>, using the formula given above.

240

</DL>

241

242

243

244

245

246

<DL>

247

<DT>Random: double gsl_ran_ugaussian_tail (const gsl_rng * <VAR>r</VAR>, double <VAR>a</VAR>)

248

249

<DT>Function: double gsl_ran_ugaussian_tail_pdf (double <VAR>x</VAR>, double <VAR>a</VAR>)

250

251

These functions compute results for the tail of a unit Gaussian

252

distribution. They are equivalent to the functions above with a standard

253

deviation of one, <VAR>sigma</VAR> = 1.

254

</DL>

255

256

257

258

259

260

<H2><A NAME="SEC290" HREF="gsl-ref_toc.html#TOC290">The Bivariate Gaussian Distribution</A></H2>

261

262

263

<DL>

264

<DT>Random: void gsl_ran_bivariate_gaussian (const gsl_rng * <VAR>r</VAR>, double <VAR>sigma_x</VAR>, double <VAR>sigma_y</VAR>, double <VAR>rho</VAR>, double * <VAR>x</VAR>, double * <VAR>y</VAR>)

265

266

267

268

269

This function generates a pair of correlated gaussian variates, with

270

mean zero, correlation coefficient <VAR>rho</VAR> and standard deviations

271

<VAR>sigma_x</VAR> and <VAR>sigma_y</VAR> in the x and y directions.

272

The probability distribution for bivariate gaussian random variates is,

273

274

275

276

277

p(x,y) dx dy = {1 \over 2 \pi \sigma_x \sigma_y \sqrt{1-\rho^2}} \exp (-(x^2/\sigma_x^2 + y^2/\sigma_y^2 - 2 \rho x y/(\sigma_x\sigma_y))/2(1-\rho^2)) dx dy

278

</PRE>

279

280

281

for x,y in the range -\infty to +\infty. The

282

correlation coefficient <VAR>rho</VAR> should lie between 1 and

283

-1.

284

</DL>

285

286

287

288

<DL>

289

<DT>Function: double gsl_ran_bivariate_gaussian_pdf (double <VAR>x</VAR>, double <VAR>y</VAR>, double <VAR>sigma_x</VAR>, double <VAR>sigma_y</VAR>, double <VAR>rho</VAR>)

290

291

This function computes the probability density p(x,y) at

292

(<VAR>x</VAR>,<VAR>y</VAR>) for a bivariate gaussian distribution with standard

293

deviations <VAR>sigma_x</VAR>, <VAR>sigma_y</VAR> and correlation coefficient

294

<VAR>rho</VAR>, using the formula given above.

295

</DL>

296

297

298

299

300

301

302

<H2><A NAME="SEC291" HREF="gsl-ref_toc.html#TOC291">The Exponential Distribution</A></H2>

303

304

<DL>

305

<DT>Random: double gsl_ran_exponential (const gsl_rng * <VAR>r</VAR>, double <VAR>mu</VAR>)

306

307

308

This function returns a random variate from the exponential distribution

309

with mean <VAR>mu</VAR>. The distribution is,

310

311

312

313

314

p(x) dx = {1 \over \mu} \exp(-x/\mu) dx

315

</PRE>

316

317

318

for

319

x >= 0.

320

</DL>

321

322

323

324

<DL>

325

<DT>Function: double gsl_ran_exponential_pdf (double <VAR>x</VAR>, double <VAR>mu</VAR>)

326

327

This function computes the probability density p(x) at <VAR>x</VAR>

328

for an exponential distribution with mean <VAR>mu</VAR>, using the formula

329

given above.

330

</DL>

331

332

333

334

335

336

<DL>

337

<DT>Function: double gsl_cdf_exponential_P (double <VAR>x</VAR>, double <VAR>mu</VAR>)

338

339

<DT>Function: double gsl_cdf_exponential_Q (double <VAR>x</VAR>, double <VAR>mu</VAR>)

340

341

<DT>Function: double gsl_cdf_exponential_Pinv (double <VAR>P</VAR>, double <VAR>mu</VAR>)

342

343

<DT>Function: double gsl_cdf_exponential_Qinv (double <VAR>Q</VAR>, double <VAR>mu</VAR>)

344

345

These functions compute the cumulative distribution functions

346

P(x), Q(x) and their inverses for the exponential

347

distribution with mean <VAR>mu</VAR>.

348

</DL>

349

350

351

352

353

<H2><A NAME="SEC292" HREF="gsl-ref_toc.html#TOC292">The Laplace Distribution</A></H2>

354

355

<DL>

356

<DT>Random: double gsl_ran_laplace (const gsl_rng * <VAR>r</VAR>, double <VAR>a</VAR>)

357

358

359

360

This function returns a random variate from the Laplace distribution

361

with width <VAR>a</VAR>. The distribution is,

362

363

364

365

366

p(x) dx = {1 \over 2 a} \exp(-|x/a|) dx

367

</PRE>

368

369

370

for -\infty < x < \infty.

371

</DL>

372

373

374

375

<DL>

376

<DT>Function: double gsl_ran_laplace_pdf (double <VAR>x</VAR>, double <VAR>a</VAR>)

377

378

This function computes the probability density p(x) at <VAR>x</VAR>

379

for a Laplace distribution with width <VAR>a</VAR>, using the formula

380

given above.

381

</DL>

382

383

384

385

386

387

<DL>

388

<DT>Function: double gsl_cdf_laplace_P (double <VAR>x</VAR>, double <VAR>a</VAR>)

389

390

<DT>Function: double gsl_cdf_laplace_Q (double <VAR>x</VAR>, double <VAR>a</VAR>)

391

392

<DT>Function: double gsl_cdf_laplace_Pinv (double <VAR>P</VAR>, double <VAR>a</VAR>)

393

394

<DT>Function: double gsl_cdf_laplace_Qinv (double <VAR>Q</VAR>, double <VAR>a</VAR>)

395

396

These functions compute the cumulative distribution functions

397

P(x), Q(x) and their inverses for the Laplace

398

distribution with width <VAR>a</VAR>.

399

</DL>

400

401

402

403

404

405

<H2><A NAME="SEC293" HREF="gsl-ref_toc.html#TOC293">The Exponential Power Distribution</A></H2>

406

407

<DL>

408

<DT>Random: double gsl_ran_exppow (const gsl_rng * <VAR>r</VAR>, double <VAR>a</VAR>, double <VAR>b</VAR>)

409

410

411

This function returns a random variate from the exponential power distribution

412

with scale parameter <VAR>a</VAR> and exponent <VAR>b</VAR>. The distribution is,

413

414

415

416

417

p(x) dx = {1 \over 2 a \Gamma(1+1/b)} \exp(-|x/a|^b) dx

418

</PRE>

419

420

421

for

422

x >= 0. For b = 1 this reduces to the Laplace

423

distribution. For b = 2 it has the same form as a gaussian

424

distribution, but with

425

a = \sqrt{2} \sigma.

426

</DL>

427

428

429

430

<DL>

431

<DT>Function: double gsl_ran_exppow_pdf (double <VAR>x</VAR>, double <VAR>a</VAR>, double <VAR>b</VAR>)

432

433

This function computes the probability density p(x) at <VAR>x</VAR>

434

for an exponential power distribution with scale parameter <VAR>a</VAR>

435

and exponent <VAR>b</VAR>, using the formula given above.

436

</DL>

437

438

439

440

441

442

443

<H2><A NAME="SEC294" HREF="gsl-ref_toc.html#TOC294">The Cauchy Distribution</A></H2>

444

445

<DL>

446

<DT>Random: double gsl_ran_cauchy (const gsl_rng * <VAR>r</VAR>, double <VAR>a</VAR>)

447

448

449

This function returns a random variate from the Cauchy distribution with

450

scale parameter <VAR>a</VAR>. The probability distribution for Cauchy

451

random variates is,

452

453

454

455

456

p(x) dx = {1 \over a\pi (1 + (x/a)^2) } dx

457

</PRE>

458

459

460

for x in the range -\infty to +\infty. The Cauchy

461

distribution is also known as the Lorentz distribution.

462

</DL>

463

464

465

466

<DL>

467

<DT>Function: double gsl_ran_cauchy_pdf (double <VAR>x</VAR>, double <VAR>a</VAR>)

468

469

This function computes the probability density p(x) at <VAR>x</VAR>

470

for a Cauchy distribution with scale parameter <VAR>a</VAR>, using the formula

471

given above.

472

</DL>

473

474

475

476

477

478

<DL>

479

<DT>Function: double gsl_cdf_cauchy_P (double <VAR>x</VAR>, double <VAR>a</VAR>)

480

481

<DT>Function: double gsl_cdf_cauchy_Q (double <VAR>x</VAR>, double <VAR>a</VAR>)

482

483

<DT>Function: double gsl_cdf_cauchy_Pinv (double <VAR>P</VAR>, double <VAR>a</VAR>)

484

485

<DT>Function: double gsl_cdf_cauchy_Qinv (double <VAR>Q</VAR>, double <VAR>a</VAR>)

486

487

These functions compute the cumulative distribution functions

488

P(x), Q(x) and their inverses for the Cauchy

489

distribution with scale parameter <VAR>a</VAR>.

490

</DL>

491

492

493

494

495

496

<H2><A NAME="SEC295" HREF="gsl-ref_toc.html#TOC295">The Rayleigh Distribution</A></H2>

497

498

<DL>

499

<DT>Random: double gsl_ran_rayleigh (const gsl_rng * <VAR>r</VAR>, double <VAR>sigma</VAR>)

500

501

502

This function returns a random variate from the Rayleigh distribution with

503

scale parameter <VAR>sigma</VAR>. The distribution is,

504

505

506

507

508

p(x) dx = {x \over \sigma^2} \exp(- x^2/(2 \sigma^2)) dx

509

</PRE>

510

511

512

for x > 0.

513

</DL>

514

515

516

517

<DL>

518

<DT>Function: double gsl_ran_rayleigh_pdf (double <VAR>x</VAR>, double <VAR>sigma</VAR>)

519

520

This function computes the probability density p(x) at <VAR>x</VAR>

521

for a Rayleigh distribution with scale parameter <VAR>sigma</VAR>, using the

522

formula given above.

523

</DL>

524

525

526

527

528

529

<DL>

530

<DT>Function: double gsl_cdf_rayleigh_P (double <VAR>x</VAR>, double <VAR>sigma</VAR>)

531

532

<DT>Function: double gsl_cdf_rayleigh_Q (double <VAR>x</VAR>, double <VAR>sigma</VAR>)

533

534

<DT>Function: double gsl_cdf_rayleigh_Pinv (double <VAR>P</VAR>, double <VAR>sigma</VAR>)

535

536

<DT>Function: double gsl_cdf_rayleigh_Qinv (double <VAR>Q</VAR>, double <VAR>sigma</VAR>)

537

538

These functions compute the cumulative distribution functions

539

P(x), Q(x) and their inverses for the Rayleigh

540

distribution with scale parameter <VAR>sigma</VAR>.

541

</DL>

542

543

544

545

546

547

<H2><A NAME="SEC296" HREF="gsl-ref_toc.html#TOC296">The Rayleigh Tail Distribution</A></H2>

548

549

<DL>

550

<DT>Random: double gsl_ran_rayleigh_tail (const gsl_rng * <VAR>r</VAR>, double <VAR>a</VAR>, double <VAR>sigma</VAR>)

551

552

553

This function returns a random variate from the tail of the Rayleigh

554

distribution with scale parameter <VAR>sigma</VAR> and a lower limit of

555

<VAR>a</VAR>. The distribution is,

556

557

558

559

560

p(x) dx = {x \over \sigma^2} \exp ((a^2 - x^2) /(2 \sigma^2)) dx

561

</PRE>

562

563

564

for x > a.

565

</DL>

566

567

568

569

<DL>

570

<DT>Function: double gsl_ran_rayleigh_tail_pdf (double <VAR>x</VAR>, double <VAR>a</VAR>, double <VAR>sigma</VAR>)

571

572

This function computes the probability density p(x) at <VAR>x</VAR>

573

for a Rayleigh tail distribution with scale parameter <VAR>sigma</VAR> and

574

lower limit <VAR>a</VAR>, using the formula given above.

575

</DL>

576

577

578

579

580

581

582

<H2><A NAME="SEC297" HREF="gsl-ref_toc.html#TOC297">The Landau Distribution</A></H2>

583

584

<DL>

585

<DT>Random: double gsl_ran_landau (const gsl_rng * <VAR>r</VAR>)

586

587

588

This function returns a random variate from the Landau distribution. The

589

probability distribution for Landau random variates is defined

590

analytically by the complex integral,

591

592

593

594

595

p(x) = (1/(2 \pi i)) \int_{c-i\infty}^{c+i\infty} ds exp(s log(s) + x s)

596

</PRE>

597

598

599

For numerical purposes it is more convenient to use the following

600

equivalent form of the integral,

601

602

603

p(x) = (1/\pi) \int_0^\infty dt \exp(-t \log(t) - x t) \sin(\pi t).

604

</PRE>

605

606

</DL>

607

608

609

<DL>

610

<DT>Function: double gsl_ran_landau_pdf (double <VAR>x</VAR>)

611

612

This function computes the probability density p(x) at <VAR>x</VAR>

613

for the Landau distribution using an approximation to the formula given

614

above.

615

</DL>

616

617

618

619

620

621

622

<H2><A NAME="SEC298" HREF="gsl-ref_toc.html#TOC298">The Levy alpha-Stable Distributions</A></H2>

623

624

<DL>

625

<DT>Random: double gsl_ran_levy (const gsl_rng * <VAR>r</VAR>, double <VAR>c</VAR>, double <VAR>alpha</VAR>)

626

627

628

This function returns a random variate from the Levy symmetric stable

629

distribution with scale <VAR>c</VAR> and exponent <VAR>alpha</VAR>. The symmetric

630

stable probability distribution is defined by a fourier transform,

631

632

633

634

635

p(x) = {1 \over 2 \pi} \int_{-\infty}^{+\infty} dt \exp(-it x - |c t|^alpha)

636

</PRE>

637

638

639

There is no explicit solution for the form of p(x) and the

640

library does not define a corresponding <CODE>pdf</CODE> function. For

641

\alpha = 1 the distribution reduces to the Cauchy distribution. For

642

\alpha = 2 it is a Gaussian distribution with

643

\sigma = \sqrt{2} c. For \alpha < 1 the tails of the

644

distribution become extremely wide.

645

646

647

648

The algorithm only works for

649

0 < alpha <= 2.

650

</DL>

651

652

653

654

655

656

657

<H2><A NAME="SEC299" HREF="gsl-ref_toc.html#TOC299">The Levy skew alpha-Stable Distribution</A></H2>

658

659

660

<DL>

661

<DT>Random: double gsl_ran_levy_skew (const gsl_rng * <VAR>r</VAR>, double <VAR>c</VAR>, double <VAR>alpha</VAR>, double <VAR>beta</VAR>)

662

663

664

665

This function returns a random variate from the Levy skew stable

666

distribution with scale <VAR>c</VAR>, exponent <VAR>alpha</VAR> and skewness

667

parameter <VAR>beta</VAR>. The skewness parameter must lie in the range

668

[-1,1]. The Levy skew stable probability distribution is defined

669

by a fourier transform,

670

671

672

673

674

p(x) = {1 \over 2 \pi} \int_{-\infty}^{+\infty} dt \exp(-it x - |c t|^alpha (1-i beta sign(t) tan(pi alpha/2)))

675

</PRE>

676

677

678

When \alpha = 1 the term \tan(\pi \alpha/2) is replaced by

679

-(2/\pi)\log|t|. There is no explicit solution for the form of

680

p(x) and the library does not define a corresponding <CODE>pdf</CODE>

681

function. For \alpha = 2 the distribution reduces to a Gaussian

682

distribution with

683

\sigma = \sqrt{2} c and the skewness parameter has no effect.

684

For \alpha < 1 the tails of the distribution become extremely

685

wide. The symmetric distribution corresponds to \beta =

686

687

688

689

690

The algorithm only works for

691

0 < alpha <= 2.

692

</DL>

693

694

695

696

The Levy alpha-stable distributions have the property that if N

697

alpha-stable variates are drawn from the distribution p(c, \alpha,

698

\beta) then the sum Y = X_1 + X_2 + \dots + X_N will also be

699

distributed as an alpha-stable variate,

700

p(N^(1/\alpha) c, \alpha, \beta).

701

702

703

704

705

706

707

708

<H2><A NAME="SEC300" HREF="gsl-ref_toc.html#TOC300">The Gamma Distribution</A></H2>

709

710

<DL>

711

<DT>Random: double gsl_ran_gamma (const gsl_rng * <VAR>r</VAR>, double <VAR>a</VAR>, double <VAR>b</VAR>)

712

713

714

This function returns a random variate from the gamma

715

distribution. The distribution function is,

716

717

718

719

720

p(x) dx = {1 \over \Gamma(a) b^a} x^{a-1} e^{-x/b} dx

721

</PRE>

722

723

724

for x > 0.

725

</DL>

726

727

728

729

<DL>

730

<DT>Function: double gsl_ran_gamma_pdf (double <VAR>x</VAR>, double <VAR>a</VAR>, double <VAR>b</VAR>)

731

732

This function computes the probability density p(x) at <VAR>x</VAR>

733

for a gamma distribution with parameters <VAR>a</VAR> and <VAR>b</VAR>, using the

734

formula given above.

735

</DL>

736

737

738

739

740

741

<DL>

742

<DT>Function: double gsl_cdf_gamma_P (double <VAR>x</VAR>, double <VAR>a</VAR>, double <VAR>b</VAR>)

743

744

<DT>Function: double gsl_cdf_gamma_Q (double <VAR>x</VAR>, double <VAR>a</VAR>, double <VAR>b</VAR>)

745

746

<DT>Function: double gsl_cdf_gamma_Pinv (double <VAR>P</VAR>, double <VAR>a</VAR>, double <VAR>b</VAR>)

747

748

<DT>Function: double gsl_cdf_gamma_Qinv (double <VAR>Q</VAR>, double <VAR>a</VAR>, double <VAR>b</VAR>)

749

750

These functions compute the cumulative distribution functions

751

P(x), Q(x) and their inverses for the gamma

752

distribution with parameters <VAR>a</VAR> and <VAR>b</VAR>.

753

</DL>

754

755

756

757

758

<H2><A NAME="SEC301" HREF="gsl-ref_toc.html#TOC301">The Flat (Uniform) Distribution</A></H2>

759

760

<DL>

761

<DT>Random: double gsl_ran_flat (const gsl_rng * <VAR>r</VAR>, double <VAR>a</VAR>, double <VAR>b</VAR>)

762

763

764

765

This function returns a random variate from the flat (uniform)

766

distribution from <VAR>a</VAR> to <VAR>b</VAR>. The distribution is,

767

768

769

770

771

p(x) dx = {1 \over (b-a)} dx

772

</PRE>

773

774

775

776

a <= x < b and 0 otherwise.

777

</DL>

778

779

780

781

<DL>

782

<DT>Function: double gsl_ran_flat_pdf (double <VAR>x</VAR>, double <VAR>a</VAR>, double <VAR>b</VAR>)

783

784

This function computes the probability density p(x) at <VAR>x</VAR>

785

for a uniform distribution from <VAR>a</VAR> to <VAR>b</VAR>, using the formula

786

given above.

787

</DL>

788

789

790

791

792

793

<DL>

794

<DT>Function: double gsl_cdf_flat_P (double <VAR>x</VAR>, double <VAR>a</VAR>, double <VAR>b</VAR>)

795

796

<DT>Function: double gsl_cdf_flat_Q (double <VAR>x</VAR>, double <VAR>a</VAR>, double <VAR>b</VAR>)

797

798

<DT>Function: double gsl_cdf_flat_Pinv (double <VAR>P</VAR>, double <VAR>a</VAR>, double <VAR>b</VAR>)

799

800

<DT>Function: double gsl_cdf_flat_Qinv (double <VAR>Q</VAR>, double <VAR>a</VAR>, double <VAR>b</VAR>)

801

802

These functions compute the cumulative distribution functions

803

P(x), Q(x) and their inverses for a uniform distribution

804

from <VAR>a</VAR> to <VAR>b</VAR>.

805

</DL>

806

807

808

809

810

811

<H2><A NAME="SEC302" HREF="gsl-ref_toc.html#TOC302">The Lognormal Distribution</A></H2>

812

813

<DL>

814

<DT>Random: double gsl_ran_lognormal (const gsl_rng * <VAR>r</VAR>, double <VAR>zeta</VAR>, double <VAR>sigma</VAR>)

815

816

817

This function returns a random variate from the lognormal

818

distribution. The distribution function is,

819

820

821

822

823

p(x) dx = {1 \over x \sqrt{2 \pi \sigma^2} } \exp(-(\ln(x) - \zeta)^2/2 \sigma^2) dx

824

</PRE>

825

826

827

for x > 0.

828

</DL>

829

830

831

832

<DL>

833

<DT>Function: double gsl_ran_lognormal_pdf (double <VAR>x</VAR>, double <VAR>zeta</VAR>, double <VAR>sigma</VAR>)

834

835

This function computes the probability density p(x) at <VAR>x</VAR>

836

for a lognormal distribution with parameters <VAR>zeta</VAR> and <VAR>sigma</VAR>,

837

using the formula given above.

838

</DL>

839

840

841

842

843

844

<DL>

845

<DT>Function: double gsl_cdf_lognormal_P (double <VAR>x</VAR>, double <VAR>zeta</VAR>, double <VAR>sigma</VAR>)

846

847

<DT>Function: double gsl_cdf_lognormal_Q (double <VAR>x</VAR>, double <VAR>zeta</VAR>, double <VAR>sigma</VAR>)

848

849

<DT>Function: double gsl_cdf_lognormal_Pinv (double <VAR>P</VAR>, double <VAR>zeta</VAR>, double <VAR>sigma</VAR>)

850

851

<DT>Function: double gsl_cdf_lognormal_Qinv (double <VAR>Q</VAR>, double <VAR>zeta</VAR>, double <VAR>sigma</VAR>)

852

853

These functions compute the cumulative distribution functions

854

P(x), Q(x) and their inverses for the lognormal

855

distribution with parameters <VAR>zeta</VAR> and <VAR>sigma</VAR>.

856

</DL>

857

858

859

860

861

862

<H2><A NAME="SEC303" HREF="gsl-ref_toc.html#TOC303">The Chi-squared Distribution</A></H2>

863

864

The chi-squared distribution arises in statistics. If Y_i are

865

n independent gaussian random variates with unit variance then the

866

sum-of-squares,

867

868

869

870

871

X_i = \sum_i Y_i^2

872

</PRE>

873

874

875

has a chi-squared distribution with n degrees of freedom.

876

877

878

879

<DL>

880

<DT>Random: double gsl_ran_chisq (const gsl_rng * <VAR>r</VAR>, double <VAR>nu</VAR>)

881

882

883

This function returns a random variate from the chi-squared distribution

884

with <VAR>nu</VAR> degrees of freedom. The distribution function is,

885

886

887

888

889

p(x) dx = {1 \over 2 \Gamma(\nu/2) } (x/2)^{\nu/2 - 1} \exp(-x/2) dx

890

</PRE>

891

892

893

for

894

x >= 0.

895

</DL>

896

897

898

899

<DL>

900

<DT>Function: double gsl_ran_chisq_pdf (double <VAR>x</VAR>, double <VAR>nu</VAR>)

901

902

This function computes the probability density p(x) at <VAR>x</VAR>

903

for a chi-squared distribution with <VAR>nu</VAR> degrees of freedom, using

904

the formula given above.

905

</DL>

906

907

908

909

910

911

<DL>

912

<DT>Function: double gsl_cdf_chisq_P (double <VAR>x</VAR>, double <VAR>nu</VAR>)

913

914

<DT>Function: double gsl_cdf_chisq_Q (double <VAR>x</VAR>, double <VAR>nu</VAR>)

915

916

<DT>Function: double gsl_cdf_chisq_Pinv (double <VAR>P</VAR>, double <VAR>nu</VAR>)

917

918

<DT>Function: double gsl_cdf_chisq_Qinv (double <VAR>Q</VAR>, double <VAR>nu</VAR>)

919

920

These functions compute the cumulative distribution functions

921

P(x), Q(x) and their inverses for the chi-squared

922

distribution with <VAR>nu</VAR> degrees of freedom.

923

</DL>

924

925

926

927

928

929

<H2><A NAME="SEC304" HREF="gsl-ref_toc.html#TOC304">The F-distribution</A></H2>

930

931

The F-distribution arises in statistics. If Y_1 and Y_2

932

are chi-squared deviates with \nu_1 and \nu_2 degrees of

933

freedom then the ratio,

934

935

936

937

938

X = { (Y_1 / \nu_1) \over (Y_2 / \nu_2) }

939

</PRE>

940

941

942

has an F-distribution F(x;\nu_1,\nu_2).

943

944

945

946

<DL>

947

<DT>Random: double gsl_ran_fdist (const gsl_rng * <VAR>r</VAR>, double <VAR>nu1</VAR>, double <VAR>nu2</VAR>)

948

949

950

This function returns a random variate from the F-distribution with degrees of freedom <VAR>nu1</VAR> and <VAR>nu2</VAR>. The distribution function is,

951

952

953

954

955

p(x) dx =

956

{ \Gamma((\nu_1 + \nu_2)/2)

957

\over \Gamma(\nu_1/2) \Gamma(\nu_2/2) }

958

\nu_1^{\nu_1/2} \nu_2^{\nu_2/2}

959

x^{\nu_1/2 - 1} (\nu_2 + \nu_1 x)^{-\nu_1/2 -\nu_2/2}

960

</PRE>

961

962

963

for

964

x >= 0.

965

</DL>

966

967

968

969

<DL>

970

<DT>Function: double gsl_ran_fdist_pdf (double <VAR>x</VAR>, double <VAR>nu1</VAR>, double <VAR>nu2</VAR>)

971

972

This function computes the probability density p(x) at <VAR>x</VAR>

973

for an F-distribution with <VAR>nu1</VAR> and <VAR>nu2</VAR> degrees of freedom,

974

using the formula given above.

975

</DL>

976

977

978

979

980

981

<DL>

982

<DT>Function: double gsl_cdf_fdist_P (double <VAR>x</VAR>, double <VAR>nu1</VAR>, double <VAR>nu2</VAR>)

983

984

<DT>Function: double gsl_cdf_fdist_Q (double <VAR>x</VAR>, double <VAR>nu1</VAR>, double <VAR>nu2</VAR>)

985

986

These functions compute the cumulative distribution functions

987

P(x) and Q(x) for the F-distribution with <VAR>nu1</VAR> and

988

<VAR>nu2</VAR> degrees of freedom.

989

</DL>

990

991

992

993

994

<H2><A NAME="SEC305" HREF="gsl-ref_toc.html#TOC305">The t-distribution</A></H2>

995

996

The t-distribution arises in statistics. If Y_1 has a normal

997

distribution and Y_2 has a chi-squared distribution with

998

\nu degrees of freedom then the ratio,

999

1000

1001

1002

1003

X = { Y_1 \over \sqrt{Y_2 / \nu} }

1004

</PRE>

1005

1006

1007

has a t-distribution t(x;\nu) with \nu degrees of freedom.

1008

1009

1010

1011

<DL>

1012

<DT>Random: double gsl_ran_tdist (const gsl_rng * <VAR>r</VAR>, double <VAR>nu</VAR>)

1013

1014

1015

1016

This function returns a random variate from the t-distribution. The

1017

distribution function is,

1018

1019

1020

1021

1022

p(x) dx = {\Gamma((\nu + 1)/2) \over \sqrt{\pi \nu} \Gamma(\nu/2)}

1023

(1 + x^2/\nu)^{-(\nu + 1)/2} dx

1024

</PRE>

1025

1026

1027

for -\infty < x < +\infty.

1028

</DL>

1029

1030

1031

1032

<DL>

1033

<DT>Function: double gsl_ran_tdist_pdf (double <VAR>x</VAR>, double <VAR>nu</VAR>)

1034

1035

This function computes the probability density p(x) at <VAR>x</VAR>

1036

for a t-distribution with <VAR>nu</VAR> degrees of freedom, using the formula

1037

given above.

1038

</DL>

1039

1040

1041

1042

1043

1044

<DL>

1045

<DT>Function: double gsl_cdf_tdist_P (double <VAR>x</VAR>, double <VAR>nu</VAR>)

1046

1047

<DT>Function: double gsl_cdf_tdist_Q (double <VAR>x</VAR>, double <VAR>nu</VAR>)

1048

1049

<DT>Function: double gsl_cdf_tdist_Pinv (double <VAR>P</VAR>, double <VAR>nu</VAR>)

1050

1051

<DT>Function: double gsl_cdf_tdist_Qinv (double <VAR>Q</VAR>, double <VAR>nu</VAR>)

1052

1053

These functions compute the cumulative distribution functions

1054

P(x), Q(x) and their inverses for the t-distribution

1055

with <VAR>nu</VAR> degrees of freedom.

1056

</DL>

1057

1058

1059

1060

1061

<H2><A NAME="SEC306" HREF="gsl-ref_toc.html#TOC306">The Beta Distribution</A></H2>

1062

1063

<DL>

1064

<DT>Random: double gsl_ran_beta (const gsl_rng * <VAR>r</VAR>, double <VAR>a</VAR>, double <VAR>b</VAR>)

1065

1066

1067

This function returns a random variate from the beta

1068

distribution. The distribution function is,

1069

1070

1071

1072

1073

p(x) dx = {\Gamma(a+b) \over \Gamma(a) \Gamma(b)} x^{a-1} (1-x)^{b-1} dx

1074

</PRE>

1075

1076

1077

for

1078

0 <= x <= 1.

1079

</DL>

1080

1081

1082

1083

<DL>

1084

<DT>Function: double gsl_ran_beta_pdf (double <VAR>x</VAR>, double <VAR>a</VAR>, double <VAR>b</VAR>)

1085

1086

This function computes the probability density p(x) at <VAR>x</VAR>

1087

for a beta distribution with parameters <VAR>a</VAR> and <VAR>b</VAR>, using the

1088

formula given above.

1089

</DL>

1090

1091

1092

1093

1094

1095

<DL>

1096

<DT>Function: double gsl_cdf_beta_P (double <VAR>x</VAR>, double <VAR>a</VAR>, double <VAR>b</VAR>)

1097

1098

<DT>Function: double gsl_cdf_beta_Q (double <VAR>x</VAR>, double <VAR>a</VAR>, double <VAR>b</VAR>)

1099

1100

These functions compute the cumulative distribution functions

1101

P(x) and Q(x) for the beta distribution with

1102

parameters <VAR>a</VAR> and <VAR>b</VAR>.

1103

</DL>

1104

1105

1106

1107

1108

<H2><A NAME="SEC307" HREF="gsl-ref_toc.html#TOC307">The Logistic Distribution</A></H2>

1109

1110

1111

<DL>

1112

<DT>Random: double gsl_ran_logistic (const gsl_rng * <VAR>r</VAR>, double <VAR>a</VAR>)

1113

1114

1115

This function returns a random variate from the logistic

1116

distribution. The distribution function is,

1117

1118

1119

1120

1121

p(x) dx = { \exp(-x/a) \over a (1 + \exp(-x/a))^2 } dx

1122

</PRE>

1123

1124

1125

for -\infty < x < +\infty.

1126

</DL>

1127

1128

1129

1130

<DL>

1131

<DT>Function: double gsl_ran_logistic_pdf (double <VAR>x</VAR>, double <VAR>a</VAR>)

1132

1133

This function computes the probability density p(x) at <VAR>x</VAR>

1134

for a logistic distribution with scale parameter <VAR>a</VAR>, using the

1135

formula given above.

1136

</DL>

1137

1138

1139

1140

1141

1142

<DL>

1143

<DT>Function: double gsl_cdf_logistic_P (double <VAR>x</VAR>, double <VAR>a</VAR>)

1144

1145

<DT>Function: double gsl_cdf_logistic_Q (double <VAR>x</VAR>, double <VAR>a</VAR>)

1146

1147

<DT>Function: double gsl_cdf_logistic_Pinv (double <VAR>P</VAR>, double <VAR>a</VAR>)

1148

1149

<DT>Function: double gsl_cdf_logistic_Qinv (double <VAR>Q</VAR>, double <VAR>a</VAR>)

1150

1151

These functions compute the cumulative distribution functions

1152

P(x), Q(x) and their inverses for the logistic

1153

distribution with scale parameter <VAR>a</VAR>.

1154

</DL>

1155

1156

1157

1158

1159

<H2><A NAME="SEC308" HREF="gsl-ref_toc.html#TOC308">The Pareto Distribution</A></H2>

1160

1161

<DL>

1162

<DT>Random: double gsl_ran_pareto (const gsl_rng * <VAR>r</VAR>, double <VAR>a</VAR>, double <VAR>b</VAR>)

1163

1164

1165

This function returns a random variate from the Pareto distribution of

1166

order <VAR>a</VAR>. The distribution function is,

1167

1168

1169

1170

1171

p(x) dx = (a/b) / (x/b)^{a+1} dx

1172

</PRE>

1173

1174

1175

for

1176

x >= b.

1177

</DL>

1178

1179

1180

1181

<DL>

1182

<DT>Function: double gsl_ran_pareto_pdf (double <VAR>x</VAR>, double <VAR>a</VAR>, double <VAR>b</VAR>)

1183

1184

This function computes the probability density p(x) at <VAR>x</VAR>

1185

for a Pareto distribution with exponent <VAR>a</VAR> and scale <VAR>b</VAR>, using

1186

the formula given above.

1187

</DL>

1188

1189

1190

1191

1192

1193

<DL>

1194

<DT>Function: double gsl_cdf_pareto_P (double <VAR>x</VAR>, double <VAR>a</VAR>, double <VAR>b</VAR>)

1195

1196

<DT>Function: double gsl_cdf_pareto_Q (double <VAR>x</VAR>, double <VAR>a</VAR>, double <VAR>b</VAR>)

1197

1198

<DT>Function: double gsl_cdf_pareto_Pinv (double <VAR>P</VAR>, double <VAR>a</VAR>, double <VAR>b</VAR>)

1199

1200

<DT>Function: double gsl_cdf_pareto_Qinv (double <VAR>Q</VAR>, double <VAR>a</VAR>, double <VAR>b</VAR>)

1201

1202

These functions compute the cumulative distribution functions

1203

P(x), Q(x) and their inverses for the Pareto

1204

distribution with exponent <VAR>a</VAR> and scale <VAR>b</VAR>.

1205

</DL>

1206

1207

1208

1209

1210

<H2><A NAME="SEC309" HREF="gsl-ref_toc.html#TOC309">Spherical Vector Distributions</A></H2>

1211

1212

1213

The spherical distributions generate random vectors, located on a

1214

spherical surface. They can be used as random directions, for example in

1215

the steps of a random walk.

1216

1217

1218

1219

<DL>

1220

<DT>Random: void gsl_ran_dir_2d (const gsl_rng * <VAR>r</VAR>, double *<VAR>x</VAR>, double *<VAR>y</VAR>)

1221

1222

<DT>Random: void gsl_ran_dir_2d_trig_method (const gsl_rng * <VAR>r</VAR>, double *<VAR>x</VAR>, double *<VAR>y</VAR>)

1223

1224

1225

1226

1227

This function returns a random direction vector v =

1228

(<VAR>x</VAR>,<VAR>y</VAR>) in two dimensions. The vector is normalized such that

1229

|v|^2 = x^2 + y^2 = 1. The obvious way to do this is to take a

1230

uniform random number between 0 and 2\pi and let <VAR>x</VAR> and

1231

<VAR>y</VAR> be the sine and cosine respectively. Two trig functions would

1232

have been expensive in the old days, but with modern hardware

1233

implementations, this is sometimes the fastest way to go. This is the

1234

case for the Pentium (but not the case for the Sun Sparcstation).

1235

One can avoid the trig evaluations by choosing <VAR>x</VAR> and

1236

<VAR>y</VAR> in the interior of a unit circle (choose them at random from the

1237

interior of the enclosing square, and then reject those that are outside

1238

the unit circle), and then dividing by

1239

\sqrt{x^2 + y^2}.

1240

A much cleverer approach, attributed to von Neumann (See Knuth, v2, 3rd

1241

ed, p140, exercise 23), requires neither trig nor a square root. In

1242

this approach, <VAR>u</VAR> and <VAR>v</VAR> are chosen at random from the

1243

interior of a unit circle, and then x=(u^2-v^2)/(u^2+v^2) and

1244

y=2uv/(u^2+v^2).

1245

</DL>

1246

1247

1248

1249

<DL>

1250

<DT>Random: void gsl_ran_dir_3d (const gsl_rng * <VAR>r</VAR>, double *<VAR>x</VAR>, double *<VAR>y</VAR>, double * <VAR>z</VAR>)

1251

1252

1253

1254

1255

This function returns a random direction vector v =

1256

(<VAR>x</VAR>,<VAR>y</VAR>,<VAR>z</VAR>) in three dimensions. The vector is normalized

1257

such that |v|^2 = x^2 + y^2 + z^2 = 1. The method employed is

1258

due to Robert E. Knop (CACM 13, 326 (1970)), and explained in Knuth, v2,

1259

3rd ed, p136. It uses the surprising fact that the distribution

1260

projected along any axis is actually uniform (this is only true for 3

1261

dimensions).

1262

</DL>

1263

1264

1265

1266

<DL>

1267

<DT>Random: void gsl_ran_dir_nd (const gsl_rng * <VAR>r</VAR>, size_t <VAR>n</VAR>, double *<VAR>x</VAR>)

1268

1269

1270

1271

1272

1273

1274

1275

This function returns a random direction vector

1276

v = (x_1,x_2,...,x_n) in <VAR>n</VAR> dimensions. The vector is normalized

1277

such that

1278

|v|^2 = x_1^2 + x_2^2 + ... + x_n^2 = 1. The method

1279

uses the fact that a multivariate gaussian distribution is spherically

1280

symmetric. Each component is generated to have a gaussian distribution,

1281

and then the components are normalized. The method is described by

1282

Knuth, v2, 3rd ed, p135-136, and attributed to G. W. Brown, Modern

1283

Mathematics for the Engineer (1956).

1284

</DL>

1285

1286

1287

1288

1289

<H2><A NAME="SEC310" HREF="gsl-ref_toc.html#TOC310">The Weibull Distribution</A></H2>

1290

1291

<DL>

1292

<DT>Random: double gsl_ran_weibull (const gsl_rng * <VAR>r</VAR>, double <VAR>a</VAR>, double <VAR>b</VAR>)

1293

1294

1295

This function returns a random variate from the Weibull distribution. The

1296

distribution function is,

1297

1298

1299

1300

1301

p(x) dx = {b \over a^b} x^{b-1} \exp(-(x/a)^b) dx

1302

</PRE>

1303

1304

1305

for

1306

x >= 0.

1307

</DL>

1308

1309

1310

1311

<DL>

1312

<DT>Function: double gsl_ran_weibull_pdf (double <VAR>x</VAR>, double <VAR>a</VAR>, double <VAR>b</VAR>)

1313

1314

This function computes the probability density p(x) at <VAR>x</VAR>

1315

for a Weibull distribution with scale <VAR>a</VAR> and exponent <VAR>b</VAR>,

1316

using the formula given above.

1317

</DL>

1318

1319

1320

1321

1322

1323

<DL>

1324

<DT>Function: double gsl_cdf_weibull_P (double <VAR>x</VAR>, double <VAR>a</VAR>, double <VAR>b</VAR>)

1325

1326

<DT>Function: double gsl_cdf_weibull_Q (double <VAR>x</VAR>, double <VAR>a</VAR>, double <VAR>b</VAR>)

1327

1328

<DT>Function: double gsl_cdf_weibull_Pinv (double <VAR>P</VAR>, double <VAR>a</VAR>, double <VAR>b</VAR>)

1329

1330

<DT>Function: double gsl_cdf_weibull_Qinv (double <VAR>Q</VAR>, double <VAR>a</VAR>, double <VAR>b</VAR>)

1331

1332

These functions compute the cumulative distribution functions

1333

P(x), Q(x) and their inverses for the Weibull

1334

distribution with scale <VAR>a</VAR> and exponent <VAR>b</VAR>.

1335

</DL>

1336

1337

1338

1339

1340

1341

<H2><A NAME="SEC311" HREF="gsl-ref_toc.html#TOC311">The Type-1 Gumbel Distribution</A></H2>

1342

1343

<DL>

1344

<DT>Random: double gsl_ran_gumbel1 (const gsl_rng * <VAR>r</VAR>, double <VAR>a</VAR>, double <VAR>b</VAR>)

1345

1346

1347

1348

This function returns a random variate from the Type-1 Gumbel

1349

distribution. The Type-1 Gumbel distribution function is,

1350

1351

1352

1353

1354

p(x) dx = a b \exp(-(b \exp(-ax) + ax)) dx

1355

</PRE>

1356

1357

1358

for -\infty < x < \infty.

1359

</DL>

1360

1361

1362

1363

<DL>

1364

<DT>Function: double gsl_ran_gumbel1_pdf (double <VAR>x</VAR>, double <VAR>a</VAR>, double <VAR>b</VAR>)

1365

1366

This function computes the probability density p(x) at <VAR>x</VAR>

1367

for a Type-1 Gumbel distribution with parameters <VAR>a</VAR> and <VAR>b</VAR>,

1368

using the formula given above.

1369

</DL>

1370

1371

1372

1373

1374

1375

<DL>

1376

<DT>Function: double gsl_cdf_gumbel1_P (double <VAR>x</VAR>, double <VAR>a</VAR>, double <VAR>b</VAR>)

1377

1378

<DT>Function: double gsl_cdf_gumbel1_Q (double <VAR>x</VAR>, double <VAR>a</VAR>, double <VAR>b</VAR>)

1379

1380

<DT>Function: double gsl_cdf_gumbel1_Pinv (double <VAR>P</VAR>, double <VAR>a</VAR>, double <VAR>b</VAR>)

1381

1382

<DT>Function: double gsl_cdf_gumbel1_Qinv (double <VAR>Q</VAR>, double <VAR>a</VAR>, double <VAR>b</VAR>)

1383

1384

These functions compute the cumulative distribution functions

1385

P(x), Q(x) and their inverses for the Type-1 Gumbel

1386

distribution with parameters <VAR>a</VAR> and <VAR>b</VAR>.

1387

</DL>

1388

1389

1390

1391

1392

1393

<H2><A NAME="SEC312" HREF="gsl-ref_toc.html#TOC312">The Type-2 Gumbel Distribution</A></H2>

1394

1395

<DL>

1396

<DT>Random: double gsl_ran_gumbel2 (const gsl_rng * <VAR>r</VAR>, double <VAR>a</VAR>, double <VAR>b</VAR>)

1397

1398

1399

1400

This function returns a random variate from the Type-2 Gumbel

1401

distribution. The Type-2 Gumbel distribution function is,

1402

1403

1404

1405

1406

p(x) dx = a b x^{-a-1} \exp(-b x^{-a}) dx

1407

</PRE>

1408

1409

1410

for 0 < x < \infty.

1411

</DL>

1412

1413

1414

1415

<DL>

1416

<DT>Function: double gsl_ran_gumbel2_pdf (double <VAR>x</VAR>, double <VAR>a</VAR>, double <VAR>b</VAR>)

1417

1418

This function computes the probability density p(x) at <VAR>x</VAR>

1419

for a Type-2 Gumbel distribution with parameters <VAR>a</VAR> and <VAR>b</VAR>,

1420

using the formula given above.

1421

</DL>

1422

1423

1424

1425

1426

1427

<DL>

1428

<DT>Function: double gsl_cdf_gumbel2_P (double <VAR>x</VAR>, double <VAR>a</VAR>, double <VAR>b</VAR>)

1429

1430

<DT>Function: double gsl_cdf_gumbel2_Q (double <VAR>x</VAR>, double <VAR>a</VAR>, double <VAR>b</VAR>)

1431

1432

<DT>Function: double gsl_cdf_gumbel2_Pinv (double <VAR>P</VAR>, double <VAR>a</VAR>, double <VAR>b</VAR>)

1433

1434

<DT>Function: double gsl_cdf_gumbel2_Qinv (double <VAR>Q</VAR>, double <VAR>a</VAR>, double <VAR>b</VAR>)

1435

1436

These functions compute the cumulative distribution functions

1437

P(x), Q(x) and their inverses for the Type-2 Gumbel

1438

distribution with parameters <VAR>a</VAR> and <VAR>b</VAR>.

1439

</DL>

1440

1441

1442

1443

1444

1445

<H2><A NAME="SEC313" HREF="gsl-ref_toc.html#TOC313">The Dirichlet Distribution</A></H2>

1446

1447

<DL>

1448

<DT>Random: void gsl_ran_dirichlet (const gsl_rng * <VAR>r</VAR>, size_t <VAR>K</VAR>, const double <VAR>alpha</VAR>[], double <VAR>theta</VAR>[])

1449

1450

1451

This function returns an array of <VAR>K</VAR> random variates from a Dirichlet

1452

distribution of order <VAR>K</VAR>-1. The distribution function is

1453

1454

1455

1456

1457

p(\theta_1, ..., \theta_K) d\theta_1 ... d\theta_K =

1458

(1/Z) \prod_{i=1}^K \theta_i^{\alpha_i - 1} \delta(1 -\sum_{i=1}^K \theta_i) d\theta_1 ... d\theta_K

1459

</PRE>

1460

1461

1462

for

1463

theta_i >= 0

1464

and

1465

alpha_i >= 0. The delta function ensures that \sum \theta_i = 1.

1466

The normalization factor Z is

1467

1468

1469

1470

1471

Z = {\prod_{i=1}^K \Gamma(\alpha_i)} / {\Gamma( \sum_{i=1}^K \alpha_i)}

1472

</PRE>

1473

1474

1475

The random variates are generated by sampling <VAR>K</VAR> values

1476

from gamma distributions with parameters

1477

a=alpha_i, b=1,

1478

and renormalizing.

1479

See A.M. Law, W.D. Kelton, <CITE>Simulation Modeling and Analysis</CITE> (1991).

1480

</DL>

1481

1482

1483

1484

<DL>

1485

<DT>Function: double gsl_ran_dirichlet_pdf (size_t <VAR>K</VAR>, const double <VAR>alpha</VAR>[], const double <VAR>theta</VAR>[])

1486

1487

This function computes the probability density

1488

p(\theta_1, ... , \theta_K)

1489

at <VAR>theta</VAR>[<VAR>K</VAR>] for a Dirichlet distribution with parameters

1490

<VAR>alpha</VAR>[<VAR>K</VAR>], using the formula given above.

1491

</DL>

1492

1493

1494

1495

<DL>

1496

<DT>Function: double gsl_ran_dirichlet_lnpdf (size_t <VAR>K</VAR>, const double <VAR>alpha</VAR>[], const double <VAR>theta</VAR>[])

1497

1498

This function computes the logarithm of the probability density

1499

p(\theta_1, ... , \theta_K)

1500

for a Dirichlet distribution with parameters

1501

<VAR>alpha</VAR>[<VAR>K</VAR>].

1502

</DL>

1503

1504

1505

1506

1507

<H2><A NAME="SEC314" HREF="gsl-ref_toc.html#TOC314">General Discrete Distributions</A></H2>

1508

1509

1510

Given K discrete events with different probabilities P[k],

1511

produce a random value k consistent with its probability.

1512

1513

1514

1515

The obvious way to do this is to preprocess the probability list by

1516

generating a cumulative probability array with K+1 elements:

1517

1518

1519

1520

1521

C[0] = 0

1522

C[k+1] = C[k]+P[k].

1523

</PRE>

1524

1525

1526

Note that this construction produces C[K]=1. Now choose a

1527

uniform deviate u between 0 and 1, and find the value of k

1528

such that

1529

C[k] <= u < C[k+1].

1530

Although this in principle requires of order \log K steps per

1531

random number generation, they are fast steps, and if you use something

1532

like \lfloor uK \rfloor as a starting point, you can often do

1533

pretty well.

1534

1535

1536

1537

But faster methods have been devised. Again, the idea is to preprocess

1538

the probability list, and save the result in some form of lookup table;

1539

then the individual calls for a random discrete event can go rapidly.

1540

An approach invented by G. Marsaglia (Generating discrete random numbers

1541

in a computer, Comm ACM 6, 37-38 (1963)) is very clever, and readers

1542

interested in examples of good algorithm design are directed to this

1543

short and well-written paper. Unfortunately, for large K,

1544

Marsaglia's lookup table can be quite large.

1545

1546

1547

1548

A much better approach is due to Alastair J. Walker (An efficient method

1549

for generating discrete random variables with general distributions, ACM

1550

Trans on Mathematical Software 3, 253-256 (1977); see also Knuth, v2,

1551

3rd ed, p120-121,139). This requires two lookup tables, one floating

1552

point and one integer, but both only of size K. After

1553

preprocessing, the random numbers are generated in O(1) time, even for

1554

large K. The preprocessing suggested by Walker requires

1555

O(K^2) effort, but that is not actually necessary, and the

1556

implementation provided here only takes O(K) effort. In general,

1557

more preprocessing leads to faster generation of the individual random

1558

numbers, but a diminishing return is reached pretty early. Knuth points

1559

out that the optimal preprocessing is combinatorially difficult for

1560

large K.

1561

1562

1563

1564

This method can be used to speed up some of the discrete random number

1565

generators below, such as the binomial distribution. To use it for

1566

something like the Poisson Distribution, a modification would have to

1567

be made, since it only takes a finite set of K outcomes.

1568

1569

1570

1571

<DL>

1572

<DT>Function: gsl_ran_discrete_t * gsl_ran_discrete_preproc (size_t <VAR>K</VAR>, const double * <VAR>P</VAR>)

1573

1574

1575

1576

This function returns a pointer to a structure that contains the lookup

1577

table for the discrete random number generator. The array <VAR>P</VAR>[] contains

1578

the probabilities of the discrete events; these array elements must all be

1579

positive, but they needn't add up to one (so you can think of them more

1580

generally as "weights")---the preprocessor will normalize appropriately.

1581

This return value is used

1582

as an argument for the <CODE>gsl_ran_discrete</CODE> function below.

1583

</DL>

1584

1585

1586

1587

<DL>

1588

<DT>Random: size_t gsl_ran_discrete (const gsl_rng * <VAR>r</VAR>, const gsl_ran_discrete_t * <VAR>g</VAR>)

1589

1590

1591

After the preprocessor, above, has been called, you use this function to

1592

get the discrete random numbers.

1593

</DL>

1594

1595

1596

1597

<DL>

1598

<DT>Function: double gsl_ran_discrete_pdf (size_t <VAR>k</VAR>, const gsl_ran_discrete_t * <VAR>g</VAR>)

1599

1600

1601

Returns the probability P[k] of observing the variable <VAR>k</VAR>.

1602

Since P[k] is not stored as part of the lookup table, it must be

1603

recomputed; this computation takes O(K), so if <VAR>K</VAR> is large

1604

and you care about the original array P[k] used to create the

1605

lookup table, then you should just keep this original array P[k]

1606

around.

1607

</DL>

1608

1609

1610

1611

<DL>

1612

<DT>Function: void gsl_ran_discrete_free (gsl_ran_discrete_t * <VAR>g</VAR>)

1613

1614

1615

De-allocates the lookup table pointed to by <VAR>g</VAR>.

1616

</DL>

1617

1618

1619

1620

1621

<H2><A NAME="SEC315" HREF="gsl-ref_toc.html#TOC315">The Poisson Distribution</A></H2>

1622

1623

<DL>

1624

<DT>Random: unsigned int gsl_ran_poisson (const gsl_rng * <VAR>r</VAR>, double <VAR>mu</VAR>)

1625

1626

1627

This function returns a random integer from the Poisson distribution

1628

with mean <VAR>mu</VAR>. The probability distribution for Poisson variates is,

1629

1630

1631

1632

1633

p(k) = {\mu^k \over k!} \exp(-\mu)

1634

</PRE>

1635

1636

1637

for

1638

k >= 0.

1639

</DL>

1640

1641

1642

1643

<DL>

1644

<DT>Function: double gsl_ran_poisson_pdf (unsigned int <VAR>k</VAR>, double <VAR>mu</VAR>)

1645

1646

This function computes the probability p(k) of obtaining <VAR>k</VAR>

1647

from a Poisson distribution with mean <VAR>mu</VAR>, using the formula

1648

given above.

1649

</DL>

1650

1651

1652

1653

1654

1655

1656

<H2><A NAME="SEC316" HREF="gsl-ref_toc.html#TOC316">The Bernoulli Distribution</A></H2>

1657

1658

<DL>

1659

<DT>Random: unsigned int gsl_ran_bernoulli (const gsl_rng * <VAR>r</VAR>, double <VAR>p</VAR>)

1660

1661

1662

This function returns either 0 or 1, the result of a Bernoulli trial

1663

with probability <VAR>p</VAR>. The probability distribution for a Bernoulli

1664

trial is,

1665

1666

1667

1668

1669

p(0) = 1 - p

1670

p(1) = p

1671

</PRE>

1672

1673

</DL>

1674

1675

1676

<DL>

1677

<DT>Function: double gsl_ran_bernoulli_pdf (unsigned int <VAR>k</VAR>, double <VAR>p</VAR>)

1678

1679

This function computes the probability p(k) of obtaining

1680

<VAR>k</VAR> from a Bernoulli distribution with probability parameter

1681

<VAR>p</VAR>, using the formula given above.

1682

</DL>

1683

1684

1685

1686

1687

1688

1689

<H2><A NAME="SEC317" HREF="gsl-ref_toc.html#TOC317">The Binomial Distribution</A></H2>

1690

1691

<DL>

1692

<DT>Random: unsigned int gsl_ran_binomial (const gsl_rng * <VAR>r</VAR>, double <VAR>p</VAR>, unsigned int <VAR>n</VAR>)

1693

1694

1695

This function returns a random integer from the binomial distribution,

1696

the number of successes in <VAR>n</VAR> independent trials with probability

1697

<VAR>p</VAR>. The probability distribution for binomial variates is,

1698

1699

1700

1701

1702

p(k) = {n! \over k! (n-k)! } p^k (1-p)^{n-k}

1703

</PRE>

1704

1705

1706

for

1707

0 <= k <= n.

1708

</DL>

1709

1710

1711

1712

<DL>

1713

<DT>Function: double gsl_ran_binomial_pdf (unsigned int <VAR>k</VAR>, double <VAR>p</VAR>, unsigned int <VAR>n</VAR>)

1714

1715

This function computes the probability p(k) of obtaining <VAR>k</VAR>

1716

from a binomial distribution with parameters <VAR>p</VAR> and <VAR>n</VAR>, using

1717

the formula given above.

1718

</DL>

1719

1720

1721

1722

1723

1724

1725

<H2><A NAME="SEC318" HREF="gsl-ref_toc.html#TOC318">The Multinomial Distribution</A></H2>

1726

1727

<DL>

1728

<DT>Random: void gsl_ran_multinomial (const gsl_rng * <VAR>r</VAR>, size_t <VAR>K</VAR>, unsigned int <VAR>N</VAR>, const double <VAR>p</VAR>[], unsigned int <VAR>n</VAR>[])

1729

1730

1731

1732

1733

1734

This function returns an array of <VAR>K</VAR> random variates from a

1735

multinomial distribution. The distribution function is,

1736

1737

1738

1739

1740

P(n_1, n_2, ..., n_K) =

1741

(N!/(n_1! n_2! ... n_K!)) p_1^n_1 p_2^n_2 ... p_K^n_K

1742

</PRE>

1743

1744

1745

where

1746

(n_1, n_2, ..., n_K)

1747

are nonnegative integers with

1748

sum_{k=1}^K n_k = N,

1749

and

1750

(p_1, p_2, ..., p_K)

1751

is a probability distribution with \sum p_i = 1.

1752

If the array <VAR>p</VAR>[<VAR>K</VAR>] is not normalized then its entries will be

1753

treated as weights and normalized appropriately.

1754

1755

1756

1757

Random variates are generated using the conditional binomial method (see

1758

C.S. David, <CITE>The computer generation of multinomial random

1759

variates</CITE>, Comp. Stat. Data Anal. 16 (1993) 205-217 for details).

1760

</DL>

1761

1762

1763

1764

<DL>

1765

<DT>Function: double gsl_ran_multinomial_pdf (size_t <VAR>K</VAR>, const double <VAR>p</VAR>[], const unsigned int <VAR>n</VAR>[])

1766

1767

This function computes the probability

1768

P(n_1, n_2, ..., n_K)

1769

of sampling <VAR>n</VAR>[<VAR>K</VAR>] from a multinomial distribution

1770

with parameters <VAR>p</VAR>[<VAR>K</VAR>], using the formula given above.

1771

</DL>

1772

1773

1774

1775

<DL>

1776

<DT>Function: double gsl_ran_multinomial_lnpdf (size_t <VAR>K</VAR>, const double <VAR>p</VAR>[], const unsigned int <VAR>n</VAR>[])

1777

1778

This function returns the logarithm of the probability for the

1779

multinomial distribution

1780

P(n_1, n_2, ..., n_K) with parameters <VAR>p</VAR>[<VAR>K</VAR>].

1781

</DL>

1782

1783

1784

1785

1786

1787

<H2><A NAME="SEC319" HREF="gsl-ref_toc.html#TOC319">The Negative Binomial Distribution</A></H2>

1788

1789

<DL>

1790

<DT>Random: unsigned int gsl_ran_negative_binomial (const gsl_rng * <VAR>r</VAR>, double <VAR>p</VAR>, double <VAR>n</VAR>)

1791

1792

1793

This function returns a random integer from the negative binomial

1794

distribution, the number of failures occurring before <VAR>n</VAR> successes

1795

in independent trials with probability <VAR>p</VAR> of success. The

1796

probability distribution for negative binomial variates is,

1797

1798

1799

1800

1801

p(k) = {\Gamma(n + k) \over \Gamma(k+1) \Gamma(n) } p^n (1-p)^k

1802

</PRE>

1803

1804

1805

Note that n is not required to be an integer.

1806

</DL>

1807

1808

1809

1810

<DL>

1811

<DT>Function: double gsl_ran_negative_binomial_pdf (unsigned int <VAR>k</VAR>, double <VAR>p</VAR>, double <VAR>n</VAR>)

1812

1813

This function computes the probability p(k) of obtaining <VAR>k</VAR>

1814

from a negative binomial distribution with parameters <VAR>p</VAR> and

1815

<VAR>n</VAR>, using the formula given above.

1816

</DL>

1817

1818

1819

1820

1821

1822

1823

<H2><A NAME="SEC320" HREF="gsl-ref_toc.html#TOC320">The Pascal Distribution</A></H2>

1824

1825

1826

<DL>

1827

<DT>Random: unsigned int gsl_ran_pascal (const gsl_rng * <VAR>r</VAR>, double <VAR>p</VAR>, unsigned int <VAR>n</VAR>)

1828

1829

This function returns a random integer from the Pascal distribution. The

1830

Pascal distribution is simply a negative binomial distribution with an

1831

integer value of n.

1832

1833

1834

1835

1836

p(k) = {(n + k - 1)! \over k! (n - 1)! } p^n (1-p)^k

1837

</PRE>

1838

1839

1840

for

1841

k >= 0

1842

</DL>

1843

1844

1845

1846

<DL>

1847

<DT>Function: double gsl_ran_pascal_pdf (unsigned int <VAR>k</VAR>, double <VAR>p</VAR>, unsigned int <VAR>n</VAR>)

1848

1849

This function computes the probability p(k) of obtaining <VAR>k</VAR>

1850

from a Pascal distribution with parameters <VAR>p</VAR> and

1851

<VAR>n</VAR>, using the formula given above.

1852

</DL>

1853

1854

1855

1856

1857

1858

1859

<H2><A NAME="SEC321" HREF="gsl-ref_toc.html#TOC321">The Geometric Distribution</A></H2>

1860

1861

<DL>

1862

<DT>Random: unsigned int gsl_ran_geometric (const gsl_rng * <VAR>r</VAR>, double <VAR>p</VAR>)

1863

1864

1865

This function returns a random integer from the geometric distribution,

1866

the number of independent trials with probability <VAR>p</VAR> until the

1867

first success. The probability distribution for geometric variates

1868

is,

1869

1870

1871

1872

1873

p(k) = p (1-p)^(k-1)

1874

</PRE>

1875

1876

1877

for

1878

k >= 1.

1879

</DL>

1880

1881

1882

1883

<DL>

1884

<DT>Function: double gsl_ran_geometric_pdf (unsigned int <VAR>k</VAR>, double <VAR>p</VAR>)

1885

1886

This function computes the probability p(k) of obtaining <VAR>k</VAR>

1887

from a geometric distribution with probability parameter <VAR>p</VAR>, using

1888

the formula given above.

1889

</DL>

1890

1891

1892

1893

1894

1895

1896

<H2><A NAME="SEC322" HREF="gsl-ref_toc.html#TOC322">The Hypergeometric Distribution</A></H2>

1897

1898

1899

<DL>

1900

<DT>Random: unsigned int gsl_ran_hypergeometric (const gsl_rng * <VAR>r</VAR>, unsigned int <VAR>n1</VAR>, unsigned int <VAR>n2</VAR>, unsigned int <VAR>t</VAR>)

1901

1902

1903

This function returns a random integer from the hypergeometric

1904

distribution. The probability distribution for hypergeometric

1905

random variates is,

1906

1907

1908

1909

1910

p(k) = C(n_1, k) C(n_2, t - k) / C(n_1 + n_2, t)

1911

</PRE>

1912

1913

1914

where C(a,b) = a!/(b!(a-b)!) and

1915

t <= n_1 + n_2. The

1916

domain of k is

1917

max(0,t-n_2), ..., min(t,n_1).

1918

</DL>

1919

1920

1921

1922

<DL>

1923

<DT>Function: double gsl_ran_hypergeometric_pdf (unsigned int <VAR>k</VAR>, unsigned int <VAR>n1</VAR>, unsigned int <VAR>n2</VAR>, unsigned int <VAR>t</VAR>)

1924

1925

This function computes the probability p(k) of obtaining <VAR>k</VAR>

1926

from a hypergeometric distribution with parameters <VAR>n1</VAR>, <VAR>n2</VAR>,

1927

<VAR>t</VAR>, using the formula given above.

1928

</DL>

1929

1930

1931

1932

1933

1934

1935

<H2><A NAME="SEC323" HREF="gsl-ref_toc.html#TOC323">The Logarithmic Distribution</A></H2>

1936

1937

<DL>

1938

<DT>Random: unsigned int gsl_ran_logarithmic (const gsl_rng * <VAR>r</VAR>, double <VAR>p</VAR>)

1939

1940

1941

This function returns a random integer from the logarithmic

1942

distribution. The probability distribution for logarithmic random variates

1943

is,

1944

1945

1946

1947

1948

p(k) = {-1 \over \log(1-p)} {(p^k \over k)}

1949

</PRE>

1950

1951

1952

for

1953

k >= 1.

1954

</DL>

1955

1956

1957

1958

<DL>

1959

<DT>Function: double gsl_ran_logarithmic_pdf (unsigned int <VAR>k</VAR>, double <VAR>p</VAR>)

1960

1961

This function computes the probability p(k) of obtaining <VAR>k</VAR>

1962

from a logarithmic distribution with probability parameter <VAR>p</VAR>,

1963

using the formula given above.

1964

</DL>

1965

1966

1967

1968

1969

1970

1971

<H2><A NAME="SEC324" HREF="gsl-ref_toc.html#TOC324">Shuffling and Sampling</A></H2>

1972

1973

1974

The following functions allow the shuffling and sampling of a set of

1975

objects. The algorithms rely on a random number generator as a source of

1976

randomness and a poor quality generator can lead to correlations in the

1977

output. In particular it is important to avoid generators with a short

1978

period. For more information see Knuth, v2, 3rd ed, Section 3.4.2,

1979

"Random Sampling and Shuffling".

1980

1981

1982

1983

<DL>

1984

<DT>Random: void gsl_ran_shuffle (const gsl_rng * <VAR>r</VAR>, void * <VAR>base</VAR>, size_t <VAR>n</VAR>, size_t <VAR>size</VAR>)

1985

1986

1987

1988

1989

This function randomly shuffles the order of <VAR>n</VAR> objects, each of

1990

size <VAR>size</VAR>, stored in the array <VAR>base</VAR>[0..<VAR>n</VAR>-1]. The

1991

output of the random number generator <VAR>r</VAR> is used to produce the

1992

permutation. The algorithm generates all possible n!

1993

permutations with equal probability, assuming a perfect source of random

1994

numbers.

1995

1996

1997

1998

The following code shows how to shuffle the numbers from 0 to 51,

1999

2000

2001

2002

2003

int a[52];

2004

2005

for (i = 0; i < 52; i++)

2006

{

2007

a[i] = i;

2008

}

2009

2010

gsl_ran_shuffle (r, a, 52, sizeof (int));

2011

</PRE>

2012

2013

</DL>

2014

2015

2016

<DL>

2017

<DT>Random: int gsl_ran_choose (const gsl_rng * <VAR>r</VAR>, void * <VAR>dest</VAR>, size_t <VAR>k</VAR>, void * <VAR>src</VAR>, size_t <VAR>n</VAR>, size_t <VAR>size</VAR>)

2018

2019

This function fills the array <VAR>dest</VAR>[k] with <VAR>k</VAR> objects taken

2020

randomly from the <VAR>n</VAR> elements of the array

2021

<VAR>src</VAR>[0..<VAR>n</VAR>-1]. The objects are each of size <VAR>size</VAR>. The

2022

output of the random number generator <VAR>r</VAR> is used to make the

2023

selection. The algorithm ensures all possible samples are equally

2024

likely, assuming a perfect source of randomness.

2025

2026

2027

2028

The objects are sampled without replacement, thus each object can

2029

only appear once in <VAR>dest</VAR>[k]. It is required that <VAR>k</VAR> be less

2030

than or equal to <CODE>n</CODE>. The objects in <VAR>dest</VAR> will be in the

2031

same relative order as those in <VAR>src</VAR>. You will need to call

2032

<CODE>gsl_ran_shuffle(r, dest, n, size)</CODE> if you want to randomize the

2033

order.

2034

2035

2036

2037

The following code shows how to select a random sample of three unique

2038

numbers from the set 0 to 99,

2039

2040

2041

2042

2043

double a[3], b[100];

2044

2045

for (i = 0; i < 100; i++)

2046

{

2047

b[i] = (double) i;

2048

}

2049

2050

gsl_ran_choose (r, a, 3, b, 100, sizeof (double));

2051

</PRE>

2052

2053

</DL>

2054

2055

2056

<DL>

2057

<DT>Random: void gsl_ran_sample (const gsl_rng * <VAR>r</VAR>, void * <VAR>dest</VAR>, size_t <VAR>k</VAR>, void * <VAR>src</VAR>, size_t <VAR>n</VAR>, size_t <VAR>size</VAR>)

2058

2059

This function is like <CODE>gsl_ran_choose</CODE> but samples <VAR>k</VAR> items

2060

from the original array of <VAR>n</VAR> items <VAR>src</VAR> with replacement, so

2061

the same object can appear more than once in the output sequence

2062

<VAR>dest</VAR>. There is no requirement that <VAR>k</VAR> be less than <VAR>n</VAR>

2063

in this case.

2064

</DL>

2065

2066

2067

2068

2069

2070

<H2><A NAME="SEC325" HREF="gsl-ref_toc.html#TOC325">Examples</A></H2>

2071

2072

2073

The following program demonstrates the use of a random number generator

2074

to produce variates from a distribution. It prints 10 samples from the

2075

Poisson distribution with a mean of 3.

2076

2077

2078

2079

2080

#include <stdio.h>

2081

#include <gsl/gsl_rng.h>

2082

#include <gsl/gsl_randist.h>

2083

2084

int

2085

main (void)

2086

{

2087

const gsl_rng_type * T;

2088

gsl_rng * r;

2089

2090

int i, n = 10;

2091

double mu = 3.0;

2092

2093

/* create a generator chosen by the

2094

environment variable GSL_RNG_TYPE */

2095

2096

gsl_rng_env_setup();

2097

2098

T = gsl_rng_default;

2099

r = gsl_rng_alloc (T);

2100

2101

/* print n random variates chosen from

2102

the poisson distribution with mean

2103

parameter mu */

2104

2105

for (i = 0; i < n; i++)

2106

{

2107

unsigned int k = gsl_ran_poisson (r, mu);

2108

printf (" %u", k);

2109

}

2110

2111

printf ("\n");

2112

return 0;

2113

}

2114

</PRE>

2115

2116

2117

If the library and header files are installed under <TT>'/usr/local'</TT>

2118

(the default location) then the program can be compiled with these

2119

options,

2120

2121

2122

2123

2124

gcc demo.c -lgsl -lgslcblas -lm

2125

</PRE>

2126

2127

2128

Here is the output of the program,

2129

2130

2131

2132

2133

$ ./a.out

2134

2 5 5 2 1 0 3 4 1 1

2135

</PRE>

2136

2137

2138

The variates depend on the seed used by the generator. The seed for the

2139

default generator type <CODE>gsl_rng_default</CODE> can be changed with the

2140

<CODE>GSL_RNG_SEED</CODE> environment variable to produce a different stream

2141

of variates,

2142

2143

2144

2145

2146

$ GSL_RNG_SEED=123 ./a.out

2147

GSL_RNG_SEED=123

2148

4 5 6 3 3 1 4 2 5 5

2149

</PRE>

2150

2151

2152

The following program generates a random walk in two dimensions.

2153

2154

2155

2156

2157

#include <stdio.h>

2158

#include <gsl/gsl_rng.h>

2159

#include <gsl/gsl_randist.h>

2160

2161

int

2162

main (void)

2163

{

2164

int i;

2165

double x = 0, y = 0, dx, dy;

2166

2167

const gsl_rng_type * T;

2168

gsl_rng * r;

2169

2170

gsl_rng_env_setup();

2171

T = gsl_rng_default;

2172

r = gsl_rng_alloc (T);

2173

2174

printf ("%g %g\n", x, y);

2175

2176

for (i = 0; i < 10; i++)

2177

{

2178

gsl_ran_dir_2d (r, &dx, &dy);

2179

x += dx; y += dy;

2180

printf ("%g %g\n", x, y);

2181

}

2182

return 0;

2183

}

2184

</PRE>

2185

2186

2187

Example output from the program, three 10-step random walks from the origin.

2188

2189

2190

2191

2192

2193

The following program computes the upper and lower cumulative

2194

distribution functions for the standard normal distribution at

2195

x=2.

2196

2197

2198

2199

2200

#include <stdio.h>

2201

#include <gsl/gsl_cdf.h>

2202

2203

int

2204

main (void)

2205

{

2206

double P, Q;

2207

double x = 2.0;

2208

2209

P = gsl_cdf_ugaussian_P (x);

2210

printf ("prob(x < %f) = %f\n", x, P);

2211

2212

Q = gsl_cdf_ugaussian_Q (x);

2213

printf ("prob(x > %f) = %f\n", x, Q);

2214

2215

x = gsl_cdf_ugaussian_Pinv (P);

2216

printf ("Pinv(%f) = %f\n", P, x);

2217

2218

x = gsl_cdf_ugaussian_Qinv (Q);

2219

printf ("Qinv(%f) = %f\n", Q, x);

2220

2221

return 0;

2222

}

2223

</PRE>

2224

2225

2226

Here is the output of the program,

2227

2228

2229

2230

2231

prob(x < 2.000000) = 0.977250

2232

prob(x > 2.000000) = 0.022750

2233

Pinv(0.977250) = 2.000000

2234

Qinv(0.022750) = 2.000000

2235

</PRE>

2236

2237

2238

2239

<H2><A NAME="SEC326" HREF="gsl-ref_toc.html#TOC326">References and Further Reading</A></H2>

2240

2241

2242

For an encyclopaedic coverage of the subject readers are advised to

2243

consult the book <CITE>Non-Uniform Random Variate Generation</CITE> by Luc

2244

Devroye. It covers every imaginable distribution and provides hundreds

2245

of algorithms.

2246

2247

2248

2249

2250

<LI>

2251

2252

Luc Devroye, <CITE>Non-Uniform Random Variate Generation</CITE>,

2253

Springer-Verlag, ISBN 0-387-96305-7.

2254

</UL>

2255

2256

2257

The subject of random variate generation is also reviewed by Knuth, who

2258

describes algorithms for all the major distributions.

2259

2260

2261

2262

2263

<LI>

2264

2265

Donald E. Knuth, <CITE>The Art of Computer Programming: Seminumerical

2266

Algorithms</CITE> (Vol 2, 3rd Ed, 1997), Addison-Wesley, ISBN 0201896842.

2267

</UL>

2268

2269

2270

The Particle Data Group provides a short review of techniques for

2271

generating distributions of random numbers in the "Monte Carlo"

2272

section of its Annual Review of Particle Physics.

2273

2274

2275

2276

2277

<LI>

2278

2279

<CITE>Review of Particle Properties</CITE>

2280

R.M. Barnett et al., Physical Review D54, 1 (1996)

2281

<A HREF="http://pdg.lbl.gov/">http://pdg.lbl.gov/</A>.

2282

</UL>

2283

2284

2285

The Review of Particle Physics is available online in postscript and pdf

2286

format.

2287

2288

2289

2290

An overview of methods used to compute cumulative distribution functions

2291

can be found in <CITE>Statistical Computing</CITE> by W.J. Kennedy and

2292

J.E. Gentle. Another general reference is <CITE>Elements of Statistical

2293

Computing</CITE> by R.A. Thisted.

2294

2295

2296

2297

2298

<LI>

2299

2300

William E. Kennedy and James E. Gentle, <CITE>Statistical Computing</CITE> (1980),

2301

Marcel Dekker, ISBN 0-8247-6898-1.

2302

</UL>

2303

2304

2305

2306

<LI>

2307

2308

Ronald A. Thisted, <CITE>Elements of Statistical Computing</CITE> (1988),

2309

Chapman & Hall, ISBN 0-412-01371-1.

2310

</UL>

2311

2312

2313

The cumulative distribution functions for the Gaussian distribution

2314

are based on the following papers,

2315

2316

2317

2318

2319

<LI>

2320

2321

<CITE>Rational Chebyshev Approximations Using Linear Equations</CITE>,

2322

W.J. Cody, W. Fraser, J.F. Hart. Numerische Mathematik 12, 242-251 (1968).

2323

</UL>

2324

2325

2326

2327

<LI>

2328

2329

<CITE>Rational Chebyshev Approximations for the Error Function</CITE>,

2330

W.J. Cody. Mathematics of Computation 23, n107, 631-637 (July 1969).

2331

</UL>

2332

2333

<HR>

2334

2335

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2336

</HTML>

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