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3<html>
4<head>
5  <meta http-equiv="Content-Language" content="en-us">
6  <meta http-equiv="Content-Type" content="text/html; charset=us-ascii">
7
8  <title>Boost Random Number Library Distributions</title>
9</head>
10
11<body bgcolor="#FFFFFF" text="#000000">
12  <h1>Random Number Library Distributions</h1>
13
14  <ul>
15    <li><a href="#intro">Introduction</a></li>
16
17    <li><a href="#synopsis">Synopsis</a></li>
18
19    <li><a href="#uniform_smallint">Class template
20    <code>uniform_smallint</code></a></li>
21
22    <li><a href="#uniform_int">Class template
23    <code>uniform_int</code></a></li>
24
25    <li><a href="#uniform_01">Class template <code>uniform_01</code></a></li>
26
27    <li><a href="#uniform_real">Class template
28    <code>uniform_real</code></a></li>
29
30    <li><a href="#bernoulli_distribution">Class template
31    <code>bernoulli_distribution</code></a></li>
32
33    <li><a href="#geometric_distribution">Class template
34    <code>geometric_distribution</code></a></li>
35
36    <li><a href="#triangle_distribution">Class template
37    <code>triangle_distribution</code></a></li>
38
39    <li><a href="#exponential_distribution">Class template
40    <code>exponential_distribution</code></a></li>
41
42    <li><a href="#normal_distribution">Class template
43    <code>normal_distribution</code></a></li>
44
45    <li><a href="#lognormal_distribution">Class template
46    <code>lognormal_distribution</code></a></li>
47
48    <li><a href="#uniform_on_sphere">Class template
49    <code>uniform_on_sphere</code></a></li>
50  </ul>
51
52  <h2><a name="intro" id="intro">Introduction</a></h2>
53
54  <p>In addition to the <a href="random-generators.html">random number
55  generators</a>, this library provides distribution functions which map one
56  distribution (often a uniform distribution provided by some generator) to
57  another.</p>
58
59  <p>Usually, there are several possible implementations of any given
60  mapping. Often, there is a choice between using more space, more
61  invocations of the underlying source of random numbers, or more
62  time-consuming arithmetic such as trigonometric functions. This interface
63  description does not mandate any specific implementation. However,
64  implementations which cannot reach certain values of the specified
65  distribution or otherwise do not converge statistically to it are not
66  acceptable.</p>
67
68  <table border="1" summary="">
69    <tr>
70      <th>distribution</th>
71
72      <th>explanation</th>
73
74      <th>example</th>
75    </tr>
76
77    <tr>
78      <td><code><a href="#uniform_smallint">uniform_smallint</a></code></td>
79
80      <td>discrete uniform distribution on a small set of integers (much
81      smaller than the range of the underlying generator)</td>
82
83      <td>drawing from an urn</td>
84    </tr>
85
86    <tr>
87      <td><code><a href="#uniform_int">uniform_int</a></code></td>
88
89      <td>discrete uniform distribution on a set of integers; the underlying
90      generator may be called several times to gather enough randomness for
91      the output</td>
92
93      <td>drawing from an urn</td>
94    </tr>
95
96    <tr>
97      <td><code><a href="#uniform_01">uniform_01</a></code></td>
98
99      <td>continuous uniform distribution on the range [0,1); important basis
100      for other distributions</td>
101
102      <td>-</td>
103    </tr>
104
105    <tr>
106      <td><code><a href="#uniform_real">uniform_real</a></code></td>
107
108      <td>continuous uniform distribution on some range [min, max) of real
109      numbers</td>
110
111      <td>for the range [0, 2pi): randomly dropping a stick and measuring its
112      angle in radiants (assuming the angle is uniformly distributed)</td>
113    </tr>
114
115    <tr>
116      <td><code><a href=
117      "#bernoulli_distribution">bernoulli_distribution</a></code></td>
118
119      <td>Bernoulli experiment: discrete boolean valued distribution with
120      configurable probability</td>
121
122      <td>tossing a coin (p=0.5)</td>
123    </tr>
124
125    <tr>
126      <td><code><a href=
127      "#geometric_distribution">geometric_distribution</a></code></td>
128
129      <td>measures distance between outcomes of repeated Bernoulli
130      experiments</td>
131
132      <td>throwing a die several times and counting the number of tries until
133      a "6" appears for the first time</td>
134    </tr>
135
136    <tr>
137      <td><code><a href=
138      "#triangle_distribution">triangle_distribution</a></code></td>
139
140      <td>?</td>
141
142      <td>?</td>
143    </tr>
144
145    <tr>
146      <td><code><a href=
147      "#exponential_distribution">exponential_distribution</a></code></td>
148
149      <td>exponential distribution</td>
150
151      <td>measuring the inter-arrival time of alpha particles emitted by
152      radioactive matter</td>
153    </tr>
154
155    <tr>
156      <td><code><a href=
157      "#normal_distribution">normal_distribution</a></code></td>
158
159      <td>counts outcomes of (infinitely) repeated Bernoulli experiments</td>
160
161      <td>tossing a coin 10000 times and counting how many front sides are
162      shown</td>
163    </tr>
164
165    <tr>
166      <td><code><a href=
167      "#lognormal_distribution">lognormal_distribution</a></code></td>
168
169      <td>lognormal distribution (sometimes used in simulations)</td>
170
171      <td>measuring the job completion time of an assembly line worker</td>
172    </tr>
173
174    <tr>
175      <td><code><a href=
176      "#uniform_on_sphere">uniform_on_sphere</a></code></td>
177
178      <td>uniform distribution on a unit sphere of arbitrary dimension</td>
179
180      <td>choosing a random point on Earth (assumed to be a sphere) where to
181      spend the next vacations</td>
182    </tr>
183  </table>
184
185  <p>The template parameters of the distribution functions are always in the
186  order</p>
187
188  <ul>
189    <li>Underlying source of random numbers</li>
190
191    <li>If applicable, return type, with a default to a reasonable type.</li>
192  </ul>
193
194  <p><em>The distribution functions no longer satisfy the input iterator
195  requirements (std:24.1.1 [lib.input.iterators]), because this is redundant
196  given the Generator interface and imposes a run-time overhead on all users.
197  Moreover, a Generator interface appeals to random number generation as
198  being more "natural". Use an <a href=
199  "../utility/iterator_adaptors.htm">iterator adaptor</a> if you need to wrap
200  any of the generators in an input iterator interface.</em></p>
201
202  <p>All of the distribution functions described below store a non-const
203  reference to the underlying source of random numbers. Therefore, the
204  distribution functions are not Assignable. However, they are
205  CopyConstructible. Copying a distribution function will copy the parameter
206  values. Furthermore, both the copy and the original will refer to the same
207  underlying source of random numbers. Therefore, both the copy and the
208  original will obtain their underlying random numbers from a single
209  sequence.</p>
210
211  <p>In this description, I have refrained from documenting those members in
212  detail which are already defined in the <a href=
213  "random-concepts.html">concept documentation</a>.</p>
214
215  <h2><a name="synopsis" id="synopsis">Synopsis of the distributions</a>
216  available from header <code>&lt;boost/random.hpp&gt;</code></h2>
217  <pre>
218namespace boost {
219  template&lt;class IntType = int&gt;
220  class uniform_smallint;
221  template&lt;class IntType = int&gt;
222  class uniform_int;
223  template&lt;class RealType = double&gt;
224  class uniform_01;
225  template&lt;class RealType = double&gt;
226  class uniform_real;
227
228  // discrete distributions
229  template&lt;class RealType = double&gt;
230  class bernoulli_distribution;
231  template&lt;class IntType = int&gt;
232  class geometric_distribution;
233
234  // continuous distributions
235  template&lt;class RealType = double&gt;
236  class triangle_distribution;
237  template&lt;class RealType = double&gt;
238  class exponential_distribution;
239  template&lt;class RealType = double&gt;
240  class normal_distribution;
241  template&lt;class RealType = double&gt;
242  class lognormal_distribution;
243  template&lt;class RealType = double,
244    class Cont = std::vector&lt;RealType&gt; &gt;
245  class uniform_on_sphere;
246}
247</pre>
248
249  <h2><a name="uniform_smallint" id="uniform_smallint">Class template
250  <code>uniform_smallint</code></a></h2>
251
252  <h3>Synopsis</h3>
253  <pre>
254#include &lt;<a href=
255"../../boost/random/uniform_smallint.hpp">boost/random/uniform_smallint.hpp</a>&gt;
256
257template&lt;class IntType = int&gt;
258class uniform_smallint
259{
260public:
261  typedef IntType input_type;
262  typedef IntType result_type;
263  static const bool has_fixed_range = false;
264  uniform_smallint(IntType min, IntType max);
265  result_type min() const;
266  result_type max() const;
267  void reset();
268  template&lt;class UniformRandomNumberGenerator&gt;
269  result_type operator()(UniformRandomNumberGenerator&amp; urng);
270};
271</pre>
272
273  <h3>Description</h3>
274
275  <p>The distribution function <code>uniform_smallint</code> models a
276  <a href="random-concepts.html#random-dist">random distribution</a>. On each
277  invocation, it returns a random integer value uniformly distributed in the
278  set of integer numbers {min, min+1, min+2, ..., max}. It assumes that the
279  desired range (max-min+1) is small compared to the range of the underlying
280  source of random numbers and thus makes no attempt to limit quantization
281  errors.</p>
282
283  <p>Let r<sub>out</sub>=(max-min+1) the desired range of integer numbers,
284  and let r<sub>base</sub> be the range of the underlying source of random
285  numbers. Then, for the uniform distribution, the theoretical probability
286  for any number i in the range r<sub>out</sub> will be p<sub>out</sub>(i) =
287  1/r<sub>out</sub>. Likewise, assume a uniform distribution on
288  r<sub>base</sub> for the underlying source of random numbers, i.e.
289  p<sub>base</sub>(i) = 1/r<sub>base</sub>. Let p<sub>out_s</sub>(i) denote
290  the random distribution generated by <code>uniform_smallint</code>. Then
291  the sum over all i in r<sub>out</sub> of
292  (p<sub>out_s</sub>(i)/p<sub>out</sub>(i) -1)<sup>2</sup> shall not exceed
293  r<sub>out</sub>/r<sub>base</sub><sup>2</sup> (r<sub>base</sub> mod
294  r<sub>out</sub>)(r<sub>out</sub> - r<sub>base</sub> mod
295  r<sub>out</sub>).</p>
296
297  <p>The template parameter <code>IntType</code> shall denote an integer-like
298  value type.</p>
299
300  <p><em>Note:</em> The property above is the square sum of the relative
301  differences in probabilities between the desired uniform distribution
302  p<sub>out</sub>(i) and the generated distribution p<sub>out_s</sub>(i). The
303  property can be fulfilled with the calculation (base_rng mod
304  r<sub>out</sub>), as follows: Let r = r<sub>base</sub> mod r<sub>out</sub>.
305  The base distribution on r<sub>base</sub> is folded onto the range
306  r<sub>out</sub>. The numbers i &lt; r have assigned (r<sub>base</sub> div
307  r<sub>out</sub>)+1 numbers of the base distribution, the rest has only
308  (r<sub>base</sub> div r<sub>out</sub>). Therefore, p<sub>out_s</sub>(i) =
309  ((r<sub>base</sub> div r<sub>out</sub>)+1) / r<sub>base</sub> for i &lt; r
310  and p<sub>out_s</sub>(i) = (r<sub>base</sub> div
311  r<sub>out</sub>)/r<sub>base</sub> otherwise. Substituting this in the above
312  sum formula leads to the desired result.</p>
313
314  <p><em>Note:</em> The upper bound for (r<sub>base</sub> mod
315  r<sub>out</sub>)(r<sub>out</sub> - r<sub>base</sub> mod r<sub>out</sub>) is
316  r<sub>out</sub><sup>2</sup>/4. Regarding the upper bound for the square sum
317  of the relative quantization error of
318  r<sub>out</sub><sup>3</sup>/(4*r<sub>base</sub><sup>2</sup>), it seems wise
319  to either choose r<sub>base</sub> so that r<sub>base</sub> &gt;
320  10*r<sub>out</sub><sup>2</sup> or ensure that r<sub>base</sub> is divisible
321  by r<sub>out</sub>.</p>
322
323  <h3>Members</h3>
324  <pre>
325uniform_smallint(IntType min, IntType max)
326</pre>
327
328  <p><strong>Effects:</strong> Constructs a <code>uniform_smallint</code>
329  functor. <code>min</code> and <code>max</code> are the lower and upper
330  bounds of the output range, respectively.</p>
331
332  <h2><a name="uniform_int" id="uniform_int">Class template
333  <code>uniform_int</code></a></h2>
334
335  <h3>Synopsis</h3>
336  <pre>
337#include &lt;<a href=
338"../../boost/random/uniform_int.hpp">boost/random/uniform_int.hpp</a>&gt;
339
340template&lt;class IntType = int&gt;
341class uniform_int
342{
343public:
344  typedef IntType input_type;
345  typedef IntType result_type;
346  static const bool has_fixed_range = false;
347  explicit uniform_int(IntType min = 0, IntType max = 9);
348  result_type min() const;
349  result_type max() const;
350  void reset();
351  template&lt;class UniformRandomNumberGenerator&gt;
352  result_type operator()(UniformRandomNumberGenerator&amp; urng);
353  template&lt;class UniformRandomNumberGenerator&gt;
354  result_type operator()(UniformRandomNumberGenerator&amp; urng, result_type n);
355};
356</pre>
357
358  <h3>Description</h3>
359
360  <p>The distribution function <code>uniform_int</code> models a <a href=
361  "random-concepts.html#random-dist">random distribution</a>. On each
362  invocation, it returns a random integer value uniformly distributed in the
363  set of integer numbers {min, min+1, min+2, ..., max}.</p>
364
365  <p>The template parameter <code>IntType</code> shall denote an integer-like
366  value type.</p>
367
368  <h3>Members</h3>
369  <pre>
370    uniform_int(IntType min = 0, IntType max = 9)
371</pre>
372
373  <p><strong>Requires:</strong> min &lt;= max<br>
374  <strong>Effects:</strong> Constructs a <code>uniform_int</code> object.
375  <code>min</code> and <code>max</code> are the parameters of the
376  distribution.</p>
377  <pre>
378    result_type min() const
379</pre>
380
381  <p><strong>Returns:</strong> The "min" parameter of the distribution.</p>
382  <pre>
383    result_type max() const
384</pre>
385
386  <p><strong>Returns:</strong> The "max" parameter of the distribution.</p>
387  <pre>
388    result_type operator()(UniformRandomNumberGenerator&amp; urng, result_type
389n)
390</pre>
391
392  <p><strong>Returns:</strong> A uniform random number x in the range 0 &lt;=
393  x &lt; n. <em>[Note: This allows a <code>variate_generator</code> object
394  with a <code>uniform_int</code> distribution to be used with
395  std::random_shuffe, see [lib.alg.random.shuffle]. ]</em></p>
396
397  <h2><a name="uniform_01" id="uniform_01">Class template
398  <code>uniform_01</code></a></h2>
399
400  <h3>Synopsis</h3>
401  <pre>
402#include &lt;<a href=
403"../../boost/random/uniform_01.hpp">boost/random/uniform_01.hpp</a>&gt;
404
405template&lt;class UniformRandomNumberGenerator, class RealType = double&gt;
406class uniform_01
407{
408public:
409  typedef UniformRandomNumberGenerator base_type;
410  typedef RealType result_type;
411  static const bool has_fixed_range = false;
412  explicit uniform_01(base_type &amp; rng);
413  result_type operator()();
414  result_type min() const;
415  result_type max() const;
416};
417</pre>
418
419  <h3>Description</h3>
420
421  <p>The distribution function <code>uniform_01</code> models a <a href=
422  "random-concepts.html#random-dist">random distribution</a>. On each
423  invocation, it returns a random floating-point value uniformly distributed
424  in the range [0..1). The value is computed using
425  <code>std::numeric_limits&lt;RealType&gt;::digits</code> random binary
426  digits, i.e. the mantissa of the floating-point value is completely filled
427  with random bits. [<em>Note:</em> Should this be configurable?]</p>
428
429  <p>The template parameter <code>RealType</code> shall denote a float-like
430  value type with support for binary operators +, -, and /. It must be large
431  enough to hold floating-point numbers of value
432  <code>rng.max()-rng.min()+1</code>.</p>
433
434  <p><code>base_type::result_type</code> must be a number-like value type, it
435  must support <code>static_cast&lt;&gt;</code> to <code>RealType</code> and
436  binary operator -.</p>
437
438  <p><em>Note:</em> The current implementation is buggy, because it may not
439  fill all of the mantissa with random bits. I'm unsure how to fill a
440  (to-be-invented) <code>boost::bigfloat</code> class with random bits
441  efficiently. It's probably time for a traits class.</p>
442
443  <h3>Members</h3>
444  <pre>
445explicit uniform_01(base_type &amp; rng)
446</pre>
447
448  <p><strong>Effects:</strong> Constructs a <code>uniform_01</code> functor
449  with the given uniform random number generator as the underlying source of
450  random numbers.</p>
451
452  <h2><a name="uniform_real" id="uniform_real">Class template
453  <code>uniform_real</code></a></h2>
454
455  <h3>Synopsis</h3>
456  <pre>
457#include &lt;<a href=
458"../../boost/random/uniform_real.hpp">boost/random/uniform_real.hpp</a>&gt;
459
460template&lt;class RealType = double&gt;
461class uniform_real
462{
463public:
464  typedef RealType input_type;
465  typedef RealType result_type;
466  static const bool has_fixed_range = false;
467  uniform_real(RealType min = RealType(0), RealType max = RealType(1));
468  result_type min() const;
469  result_type max() const;
470  void reset();
471  template&lt;class UniformRandomNumberGenerator&gt;
472  result_type operator()(UniformRandomNumberGenerator&amp; urng);
473};
474</pre>
475
476  <h3>Description</h3>
477
478  <p>The distribution function <code>uniform_real</code> models a <a href=
479  "random-concepts.html#random-dist">random distribution</a>. On each
480  invocation, it returns a random floating-point value uniformly distributed
481  in the range [min..max). The value is computed using
482  <code>std::numeric_limits&lt;RealType&gt;::digits</code> random binary
483  digits, i.e. the mantissa of the floating-point value is completely filled
484  with random bits.</p>
485
486  <p><em>Note:</em> The current implementation is buggy, because it may not
487  fill all of the mantissa with random bits.</p>
488
489  <h3>Members</h3>
490  <pre>
491    uniform_real(RealType min = RealType(0), RealType max = RealType(1))
492</pre>
493
494  <p><strong>Requires:</strong> min &lt;= max<br>
495  <strong>Effects:</strong> Constructs a <code>uniform_real</code> object;
496  <code>min</code> and <code>max</code> are the parameters of the
497  distribution.</p>
498  <pre>
499    result_type min() const
500</pre>
501
502  <p><strong>Returns:</strong> The "min" parameter of the distribution.</p>
503  <pre>
504    result_type max() const
505</pre>
506
507  <p><strong>Returns:</strong> The "max" parameter of the distribution.</p>
508
509  <h2><a name="bernoulli_distribution" id="bernoulli_distribution">Class
510  template <code>bernoulli_distribution</code></a></h2>
511
512  <h3>Synopsis</h3>
513  <pre>
514#include &lt;<a href=
515"../../boost/random/bernoulli_distribution.hpp">boost/random/bernoulli_distribution.hpp</a>&gt;
516
517template&lt;class RealType = double&gt;
518class bernoulli_distribution
519{
520public:
521  typedef int input_type;
522  typedef bool result_type;
523
524  explicit bernoulli_distribution(const RealType&amp; p = RealType(0.5));
525  RealType p() const;
526  void reset();
527  template&lt;class UniformRandomNumberGenerator&gt;
528  result_type operator()(UniformRandomNumberGenerator&amp; urng);
529};
530</pre>
531
532  <h3>Description</h3>
533
534  <p>Instantiations of class template <code>bernoulli_distribution</code>
535  model a <a href="random-concepts.html#random-dist">random distribution</a>.
536  Such a random distribution produces <code>bool</code> values distributed
537  with probabilities P(true) = p and P(false) = 1-p. p is the parameter of
538  the distribution.</p>
539
540  <h3>Members</h3>
541  <pre>
542    bernoulli_distribution(const RealType&amp; p = RealType(0.5))
543</pre>
544
545  <p><strong>Requires:</strong> 0 &lt;= p &lt;= 1<br>
546  <strong>Effects:</strong> Constructs a <code>bernoulli_distribution</code>
547  object. <code>p</code> is the parameter of the distribution.</p>
548  <pre>
549    RealType p() const
550</pre>
551
552  <p><strong>Returns:</strong> The "p" parameter of the distribution.</p>
553
554  <h2><a name="geometric_distribution" id="geometric_distribution">Class
555  template <code>geometric_distribution</code></a></h2>
556
557  <h3>Synopsis</h3>
558  <pre>
559#include &lt;<a href=
560"../../boost/random/geometric_distribution.hpp">boost/random/geometric_distribution.hpp</a>&gt;
561
562template&lt;class UniformRandomNumberGenerator, class IntType = int&gt;
563class geometric_distribution
564{
565public:
566  typedef RealType input_type;
567  typedef IntType result_type;
568
569  explicit geometric_distribution(const RealType&amp; p = RealType(0.5));
570  RealType p() const;
571  void reset();
572  template&lt;class UniformRandomNumberGenerator&gt;
573  result_type operator()(UniformRandomNumberGenerator&amp; urng);
574};
575</pre>
576
577  <h3>Description</h3>
578
579  <p>Instantiations of class template <code>geometric_distribution</code>
580  model a <a href="random-concepts.html#random-dist">random distribution</a>.
581  A <code>geometric_distribution</code> random distribution produces integer
582  values <em>i</em> &gt;= 1 with p(i) = (1-p) * p<sup>i-1</sup>. p is the
583  parameter of the distribution.</p>
584
585  <h3>Members</h3>
586  <pre>
587    geometric_distribution(const RealType&amp; p = RealType(0.5))
588</pre>
589
590  <p><strong>Requires:</strong> 0 &lt; p &lt; 1<br>
591  <strong>Effects:</strong> Constructs a <code>geometric_distribution</code>
592  object; <code>p</code> is the parameter of the distribution.</p>
593  <pre>
594   RealType p() const
595</pre>
596
597  <p><strong>Returns:</strong> The "p" parameter of the distribution.</p>
598
599  <h2><a name="triangle_distribution" id="triangle_distribution">Class
600  template <code>triangle_distribution</code></a></h2>
601
602  <h3>Synopsis</h3>
603  <pre>
604#include &lt;<a href=
605"../../boost/random/triangle_distribution.hpp">boost/random/triangle_distribution.hpp</a>&gt;
606
607template&lt;class RealType = double&gt;
608class triangle_distribution
609{
610public:
611  typedef RealType input_type;
612  typedef RealType result_type;
613  triangle_distribution(result_type a, result_type b, result_type c);
614  result_type a() const;
615  result_type b() const;
616  result_type c() const;
617  void reset();
618  template&lt;class UniformRandomNumberGenerator&gt;
619  result_type operator()(UniformRandomNumberGenerator&amp; urng);
620};
621</pre>
622
623  <h3>Description</h3>
624
625  <p>Instantiations of class template <code>triangle_distribution</code>
626  model a <a href="random-concepts.html#random-dist">random distribution</a>.
627  The returned floating-point values <code>x</code> satisfy <code>a &lt;= x
628  &lt;= c</code>; <code>x</code> has a triangle distribution, where
629  <code>b</code> is the most probable value for <code>x</code>.</p>
630
631  <h3>Members</h3>
632  <pre>
633triangle_distribution(result_type a, result_type b, result_type c)
634</pre>
635
636  <p><strong>Effects:</strong> Constructs a
637  <code>triangle_distribution</code> functor. <code>a, b, c</code> are the
638  parameters for the distribution.</p>
639
640  <h2><a name="exponential_distribution" id="exponential_distribution">Class
641  template <code>exponential_distribution</code></a></h2>
642
643  <h3>Synopsis</h3>
644  <pre>
645#include &lt;<a href=
646"../../boost/random/exponential_distribution.hpp">boost/random/exponential_distribution.hpp</a>&gt;
647
648template&lt;class RealType = double&gt;
649class exponential_distribution
650{
651public:
652  typedef RealType input_type;
653  typedef RealType result_type;
654  explicit exponential_distribution(const result_type&amp; lambda);
655  RealType lambda() const;
656  void reset();
657  template&lt;class UniformRandomNumberGenerator&gt;
658  result_type operator()(UniformRandomNumberGenerator&amp; urng);
659};
660</pre>
661
662  <h3>Description</h3>
663
664  <p>Instantiations of class template <code>exponential_distribution</code>
665  model a <a href="random-concepts.html#random-dist">random distribution</a>.
666  Such a distribution produces random numbers x &gt; 0 distributed with
667  probability density function p(x) = lambda * exp(-lambda * x), where lambda
668  is the parameter of the distribution.</p>
669
670  <h3>Members</h3>
671  <pre>
672    exponential_distribution(const result_type&amp; lambda = result_type(1))
673</pre>
674
675  <p><strong>Requires:</strong> lambda &gt; 0<br>
676  <strong>Effects:</strong> Constructs an
677  <code>exponential_distribution</code> object with <code>rng</code> as the
678  reference to the underlying source of random numbers. <code>lambda</code>
679  is the parameter for the distribution.</p>
680  <pre>
681    RealType lambda() const
682</pre>
683
684  <p><strong>Returns:</strong> The "lambda" parameter of the
685  distribution.</p>
686
687  <h2><a name="normal_distribution" id="normal_distribution">Class template
688  <code>normal_distribution</code></a></h2>
689
690  <h3>Synopsis</h3>
691  <pre>
692#include &lt;<a href=
693"../../boost/random/normal_distribution.hpp">boost/random/normal_distribution.hpp</a>&gt;
694
695template&lt;class RealType = double&gt;
696class normal_distribution
697{
698public:
699  typedef RealType input_type;
700  typedef RealType result_type;
701  explicit normal_distribution(const result_type&amp; mean = 0,
702                               const result_type&amp; sigma = 1);
703  RealType mean() const;
704  RealType sigma() const;
705  void reset();
706  template&lt;class UniformRandomNumberGenerator&gt;
707  result_type operator()(UniformRandomNumberGenerator&amp; urng);
708};
709</pre>
710
711  <h3>Description</h3>
712
713  <p>Instantiations of class template <code>normal_distribution</code> model
714  a <a href="random-concepts.html#random-dist">random distribution</a>. Such
715  a distribution produces random numbers x distributed with probability
716  density function p(x) = 1/sqrt(2*pi*sigma) * exp(- (x-mean)<sup>2</sup> /
717  (2*sigma<sup>2</sup>) ), where mean and sigma are the parameters of the
718  distribution.</p>
719
720  <h3>Members</h3>
721  <pre>
722    explicit normal_distribution(const result_type&amp; mean = 0,
723                                 const result_type&amp; sigma = 1);
724</pre>
725
726  <p><strong>Requires:</strong> sigma &gt; 0<br>
727  <strong>Effects:</strong> Constructs a <code>normal_distribution</code>
728  object; <code>mean</code> and <code>sigma</code> are the parameters for the
729  distribution.</p>
730  <pre>
731    RealType mean() const
732</pre>
733
734  <p><strong>Returns:</strong> The "mean" parameter of the distribution.</p>
735  <pre>
736    RealType sigma() const
737</pre>
738
739  <p><strong>Returns:</strong> The "sigma" parameter of the distribution.</p>
740
741  <h2><a name="lognormal_distribution" id="lognormal_distribution">Class
742  template <code>lognormal_distribution</code></a></h2>
743
744  <h3>Synopsis</h3>
745  <pre>
746#include &lt;<a href=
747"../../boost/random/lognormal_distribution.hpp">boost/random/lognormal_distribution.hpp</a>&gt;
748
749template&lt;class RealType = double&gt;
750class lognormal_distribution
751{
752public:
753  typedef typename normal_distribution&lt;RealType&gt;::input_type
754  typedef RealType result_type;
755  explicit lognormal_distribution(const result_type&amp; mean = 1.0,
756                                  const result_type&amp; sigma = 1.0);
757  RealType&amp; mean() const;
758  RealType&amp; sigma() const;                                 
759  void reset();
760  template&lt;class UniformRandomNumberGenerator&gt;
761  result_type operator()(UniformRandomNumberGenerator&amp; urng);
762};
763</pre>
764
765  <h3>Description</h3>
766
767  <p>Instantiations of class template <code>lognormal_distribution</code>
768  model a <a href="random-concepts.html#random-dist">random distribution</a>.
769  Such a distribution produces random numbers with p(x) = 1/(x * normal_sigma
770  * sqrt(2*pi)) * exp( -(log(x)-normal_mean)<sup>2</sup> /
771  (2*normal_sigma<sup>2</sup>) ) for x &gt; 0, where normal_mean =
772  log(mean<sup>2</sup>/sqrt(sigma<sup>2</sup> + mean<sup>2</sup>)) and
773  normal_sigma = sqrt(log(1 + sigma<sup>2</sup>/mean<sup>2</sup>)).</p>
774
775  <h3>Members</h3>
776  <pre>
777lognormal_distribution(const result_type&amp; mean,
778                       const result_type&amp; sigma)
779</pre>
780
781  <p><strong>Effects:</strong> Constructs a
782  <code>lognormal_distribution</code> functor. <code>mean</code> and
783  <code>sigma</code> are the mean and standard deviation of the lognormal
784  distribution.</p>
785
786  <h2><a name="uniform_on_sphere" id="uniform_on_sphere">Class template
787  <code>uniform_on_sphere</code></a></h2>
788
789  <h3>Synopsis</h3>
790  <pre>
791#include &lt;<a href=
792"../../boost/random/uniform_on_sphere.hpp">boost/random/uniform_on_sphere.hpp</a>&gt;
793
794template&lt;class RealType = double,
795  class Cont = std::vector&lt;RealType&gt; &gt;
796class uniform_on_sphere
797{
798public:
799  typedef RealType input_type;
800  typedef Cont result_type;
801  explicit uniform_on_sphere(int dim = 2);
802  void reset();
803  template&lt;class UniformRandomNumberGenerator&gt;
804  const result_type &amp; operator()(UniformRandomNumberGenerator&amp; urng);
805};
806</pre>
807
808  <h3>Description</h3>
809
810  <p>Instantiations of class template <code>uniform_on_sphere</code> model a
811  <a href="random-concepts.html#random-dist">random distribution</a>. Such a
812  distribution produces random numbers uniformly distributed on the unit
813  sphere of arbitrary dimension <code>dim</code>. The <code>Cont</code>
814  template parameter must be a STL-like container type with
815  <code>begin</code> and <code>end</code> operations returning non-const
816  ForwardIterators of type <code>Cont::iterator</code>.</p>
817
818  <h3>Members</h3>
819  <pre>
820explicit uniform_on_sphere(int dim = 2)
821</pre>
822
823  <p><strong>Effects:</strong> Constructs a <code>uniform_on_sphere</code>
824  functor. <code>dim</code> is the dimension of the sphere.</p>
825  <hr>
826
827  <p><a href="http://validator.w3.org/check?uri=referer"><img border="0" src=
828  "http://www.w3.org/Icons/valid-html401" alt="Valid HTML 4.01 Transitional"
829  height="31" width="88"></a></p>
830
831  <p>Revised
832  <!--webbot bot="Timestamp" s-type="EDITED" s-format="%d %B, %Y" startspan -->05
833  December, 2006<!--webbot bot="Timestamp" endspan i-checksum="38516" --></p>
834
835  <p><i>Copyright &copy; 2000-2004 <a href=
836  "../../people/jens_maurer.htm">Jens Maurer</a></i></p>
837
838  <p><i>Distributed under the Boost Software License, Version 1.0. (See
839  accompanying file <a href="../../LICENSE_1_0.txt">LICENSE_1_0.txt</a> or
840  copy at <a href=
841  "http://www.boost.org/LICENSE_1_0.txt">http://www.boost.org/LICENSE_1_0.txt</a>)</i></p>
842</body>
843</html>
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