File: random-distributions.html

package info (click to toggle)
boost 1.33.1-10
links: PTS
area: main
in suites: etch, etch-m68k
size: 100,948 kB
ctags: 145,103
sloc: cpp: 573,492; xml: 49,055; python: 15,626; ansic: 13,588; sh: 2,099; yacc: 858; makefile: 660; perl: 427; lex: 111; csh: 6
file content (796 lines) | stat: -rw-r--r-- 26,361 bytes
parent folder | download | duplicates (2)

<html>

<head>
<meta http-equiv="Content-Type" content="text/html; charset=iso-8859-1">

<title>Boost Random Number Library Distributions</title>
</head>

<body bgcolor="#FFFFFF" text="#000000">

<h1>Random Number Library Distributions</h1>

<ul>
<li><a href="#intro">Introduction</a>
<li><a href="#synopsis">Synopsis</a>
<li><a href="#uniform_smallint">Class template
<code>uniform_smallint</code></a>
<li><a href="#uniform_int">Class template <code>uniform_int</code></a>
<li><a href="#uniform_01">Class template <code>uniform_01</code></a>
<li><a href="#uniform_real">Class template
<code>uniform_real</code></a>
<li><a href="#bernoulli_distribution">Class template
<code>bernoulli_distribution</code></a>
<li><a href="#geometric_distribution">Class template
<code>geometric_distribution</code></a>
<li><a href="#triangle_distribution">Class template
<code>triangle_distribution</code></a>
<li><a href="#exponential_distribution">Class template
<code>exponential_distribution</code></a>
<li><a href="#normal_distribution">Class template
<code>normal_distribution</code></a>
<li><a href="#lognormal_distribution">Class template
<code>lognormal_distribution</code></a>
<li><a href="#uniform_on_sphere">Class template
<code>uniform_on_sphere</code></a>
</ul>

<h2><a name="intro">Introduction</a></h2>

In addition to the <a href="random-generators.html">random number
generators</a>, this library provides distribution functions which map
one distribution (often a uniform distribution provided by some
generator) to another.

<p>
Usually, there are several possible implementations of any given
mapping.  Often, there is a choice between using more space, more
invocations of the underlying source of random numbers, or more
time-consuming arithmetic such as trigonometric functions.  This
interface description does not mandate any specific implementation.
However, implementations which cannot reach certain values of the
specified distribution or otherwise do not converge statistically to
it are not acceptable.

<p>
<table border="1">
<tr><th>distribution</th><th>explanation</th><th>example</th></tr>

<tr>
<td><code><a href="#uniform_smallint">uniform_smallint</a></code></td>
<td>discrete uniform distribution on a small set of integers (much
smaller than the range of the underlying generator)</td>
<td>drawing from an urn</td>
</tr>

<tr>
<td><code><a href="#uniform_int">uniform_int</a></code></td>
<td>discrete uniform distribution on a set of integers; the underlying
generator may be called several times to gather enough randomness for
the output</td>
<td>drawing from an urn</td>
</tr>

<tr>
<td><code><a href="#uniform_01">uniform_01</a></code></td>
<td>continuous uniform distribution on the range [0,1); important
basis for other distributions</td>
<td>-</td>
</tr>

<tr>
<td><code><a href="#uniform_real">uniform_real</a></code></td>
<td>continuous uniform distribution on some range [min, max) of real
numbers</td>
<td>for the range [0, 2pi): randomly dropping a stick and measuring
its angle in radiants (assuming the angle is uniformly
distributed)</td>
</tr>

<tr>
<td><code><a href="#bernoulli_distribution">bernoulli_distribution</a></code></td>
<td>Bernoulli experiment: discrete boolean valued distribution with
configurable probability</td>
<td>tossing a coin (p=0.5)</td>
</tr>

<tr>
<td><code><a href="#geometric_distribution">geometric_distribution</a></code></td>
<td>measures distance between outcomes of repeated Bernoulli experiments</td>
<td>throwing a die several times and counting the number of tries
until a "6" appears for the first time</td>
</tr>

<tr>
<td><code><a href="#triangle_distribution">triangle_distribution</a></code></td>
<td>?</td>
<td>?</td>
</tr>

<tr>
<td><code><a href="#exponential_distribution">exponential_distribution</a></code></td>
<td>exponential distribution</td>
<td>measuring the inter-arrival time of alpha particles emitted by
radioactive matter</td>
</tr>

<tr>
<td><code><a href="#normal_distribution">normal_distribution</a></code></td>
<td>counts outcomes of (infinitely) repeated Bernoulli experiments</td>
<td>tossing a coin 10000 times and counting how many front sides are shown</td>
</tr>

<tr>
<td><code><a href="#lognormal_distribution">lognormal_distribution</a></code></td>
<td>lognormal distribution (sometimes used in simulations)</td>
<td>measuring the job completion time of an assembly line worker</td>
</tr>

<tr>
<td><code><a href="#uniform_on_sphere">uniform_on_sphere</a></code></td>
<td>uniform distribution on a unit sphere of arbitrary dimension</td>
<td>choosing a random point on Earth (assumed to be a sphere) where to
spend the next vacations</td>
</tr>

</table>

<p>

The template parameters of the distribution functions are always in
the order
<ul>
<li>Underlying source of random numbers
<li>If applicable, return type, with a default to a reasonable type.
</ul>

<p>
<em>The distribution functions no longer satisfy the input iterator
requirements (std:24.1.1 [lib.input.iterators]), because this is
redundant given the Generator interface and imposes a run-time
overhead on all users.  Moreover, a Generator interface appeals to
random number generation as being more "natural".  Use an
<a href="../utility/iterator_adaptors.htm">iterator adaptor</a>
if you need to wrap any of the generators in an input iterator
interface.</em>
<p>

All of the distribution functions described below store a non-const
reference to the underlying source of random numbers.  Therefore, the
distribution functions are not Assignable.  However, they are
CopyConstructible.  Copying a distribution function will copy the
parameter values.  Furthermore, both the copy and the original will
refer to the same underlying source of random numbers.  Therefore,
both the copy and the original will obtain their underlying random
numbers from a single sequence.

<p>
In this description, I have refrained from documenting those members
in detail which are already defined in the
<a href="random-concepts.html">concept documentation</a>.


<h2><a name="synopsis">Synopsis of the distributions</a> available from header
<code>&lt;boost/random.hpp&gt;</code> </h2>

<pre>
namespace boost {
  template&lt;class IntType = int&gt;
  class uniform_smallint;
  template&lt;class IntType = int&gt;
  class uniform_int;
  template&lt;class RealType = double&gt;
  class uniform_01;
  template&lt;class RealType = double&gt;
  class uniform_real;

  // discrete distributions
  template&lt;class RealType = double&gt;
  class bernoulli_distribution;
  template&lt;class IntType = int&gt;
  class geometric_distribution;

  // continuous distributions
  template&lt;class RealType = double&gt;
  class triangle_distribution;
  template&lt;class RealType = double&gt;
  class exponential_distribution;
  template&lt;class RealType = double&gt;
  class normal_distribution;
  template&lt;class RealType = double&gt;
  class lognormal_distribution;
  template&lt;class RealType = double,
    class Cont = std::vector&lt;RealType&gt; &gt;
  class uniform_on_sphere;
}
</pre>

<h2><a name="uniform_smallint">Class template
<code>uniform_smallint</code></a></h2>

<h3>Synopsis</h3>

<pre>
#include &lt;<a href="../../boost/random/uniform_smallint.hpp">boost/random/uniform_smallint.hpp</a>&gt;

template&lt;class IntType = int&gt;
class uniform_smallint
{
public:
  typedef IntType input_type;
  typedef IntType result_type;
  static const bool has_fixed_range = false;
  uniform_smallint(IntType min, IntType max);
  result_type min() const;
  result_type max() const;
  void reset();
  template&lt;class UniformRandomNumberGenerator&gt;
  result_type operator()(UniformRandomNumberGenerator& urng);
};
</pre>

<h3>Description</h3>

The distribution function <code>uniform_smallint</code> models a
<a href="random-concepts.html#random-dist">random distribution</a>.
On each invocation, it returns a random integer value
uniformly distributed in the set of integer numbers {min, min+1,
min+2, ..., max}.  It assumes that the desired range (max-min+1) is
small compared to the range of the underlying source of random
numbers and thus makes no attempt to limit quantization errors.
<p>
Let r<sub>out</sub>=(max-min+1) the desired range of integer numbers,
and let r<sub>base</sub> be the range of the underlying source of
random numbers.  Then, for the uniform distribution, the theoretical
probability for any number i in the range r<sub>out</sub> will be
p<sub>out</sub>(i) = 1/r<sub>out</sub>.  Likewise, assume a uniform
distribution on r<sub>base</sub> for the underlying source of random
numbers, i.e. p<sub>base</sub>(i) = 1/r<sub>base</sub>.  Let
p<sub>out_s</sub>(i) denote the random distribution generated by
<code>uniform_smallint</code>.  Then the sum over all i in
r<sub>out</sub> of (p<sub>out_s</sub>(i)/p<sub>out</sub>(i)
-1)<sup>2</sup> shall not exceed
r<sub>out</sub>/r<sub>base</sub><sup>2</sup> (r<sub>base</sub> mod
r<sub>out</sub>)(r<sub>out</sub> - r<sub>base</sub> mod
r<sub>out</sub>).
<p>

The template parameter <code>IntType</code> shall denote an
integer-like value type.

<p>
<em>Note:</em> The property above is the square sum of the relative
differences in probabilities between the desired uniform distribution
p<sub>out</sub>(i) and the generated distribution
p<sub>out_s</sub>(i).  The property can be fulfilled with the
calculation (base_rng mod r<sub>out</sub>), as follows: Let r =
r<sub>base</sub> mod r<sub>out</sub>.  The base distribution on
r<sub>base</sub> is folded onto the range r<sub>out</sub>.  The
numbers i &lt; r have assigned (r<sub>base</sub> div
r<sub>out</sub>)+1 numbers of the base distribution, the rest has only
(r<sub>base</sub> div r<sub>out</sub>).  Therefore,
p<sub>out_s</sub>(i) = ((r<sub>base</sub> div r<sub>out</sub>)+1) /
r<sub>base</sub> for i &lt; r and p<sub>out_s</sub>(i) =
(r<sub>base</sub> div r<sub>out</sub>)/r<sub>base</sub> otherwise.
Substituting this in the above sum formula leads to the desired
result.
<p>
<em>Note:</em> The upper bound for (r<sub>base</sub> mod r<sub>out</sub>)(r<sub>out</sub> - r<sub>base</sub>
mod r<sub>out</sub>) is r<sub>out</sub><sup>2</sup>/4.  Regarding the upper bound for the square
sum of the relative quantization error of r<sub>out</sub><sup>3</sup>/(4*r<sub>base</sub><sup>2</sup>), it
seems wise to either choose r<sub>base</sub> so that r<sub>base</sub> &gt; 10*r<sub>out</sub><sup>2</sup> or
ensure that r<sub>base</sub> is divisible by r<sub>out</sub>.


<h3>Members</h3>

<pre>uniform_smallint(IntType min, IntType max)</pre>

<strong>Effects:</strong> Constructs a <code>uniform_smallint</code>
functor. <code>min</code> and <code>max</code> are the lower and upper
bounds of the output range, respectively.


<h2><a name="uniform_int">Class template <code>uniform_int</code></a></h2>

<h3>Synopsis</h3>

<pre>
#include &lt;<a href="../../boost/random/uniform_int.hpp">boost/random/uniform_int.hpp</a>&gt;

template&lt;class IntType = int&gt;
class uniform_int
{
public:
  typedef IntType input_type;
  typedef IntType result_type;
  static const bool has_fixed_range = false;
  explicit uniform_int(IntType min = 0, IntType max = 9);
  result_type min() const;
  result_type max() const;
  void reset();
  template&lt;class UniformRandomNumberGenerator&gt;
  result_type operator()(UniformRandomNumberGenerator& urng);
  template&lt;class UniformRandomNumberGenerator&gt;
  result_type operator()(UniformRandomNumberGenerator& urng, result_type n);
};
</pre>

<h3>Description</h3>

The distribution function <code>uniform_int</code> models a
<a href="random-concepts.html#random-dist">random distribution</a>.
On each invocation, it returns a random integer
value uniformly distributed in the set of integer numbers
{min, min+1, min+2, ..., max}.
<p>

The template parameter <code>IntType</code> shall denote an
integer-like value type.

<h3>Members</h3>

<pre>    uniform_int(IntType min = 0, IntType max = 9)</pre>
<strong>Requires:</strong> min &lt;= max
<br>
<strong>Effects:</strong> Constructs a <code>uniform_int</code>
object.  <code>min</code> and <code>max</code> are the parameters of
the distribution.

<pre>    result_type min() const</pre>
<strong>Returns:</strong> The "min" parameter of the distribution.

<pre>    result_type max() const</pre>
<strong>Returns:</strong> The "max" parameter of the distribution.

<pre>    result_type operator()(UniformRandomNumberGenerator& urng, result_type 
n)</pre>
<strong>Returns:</strong> A uniform random number x in the range 0
&lt;= x &lt; n.  <em>[Note: This allows a
<code>variate_generator</code> object with a <code>uniform_int</code>
distribution to be used with std::random_shuffe, see
[lib.alg.random.shuffle]. ]</em>


<h2><a name="uniform_01">Class template <code>uniform_01</code></a></h2>

<h3>Synopsis</h3>

<pre>
#include &lt;<a href="../../boost/random/uniform_01.hpp">boost/random/uniform_01.hpp</a>&gt;

template&lt;class UniformRandomNumberGenerator, class RealType = double&gt;
class uniform_01
{
public:
  typedef UniformRandomNumberGenerator base_type;
  typedef RealType result_type;
  static const bool has_fixed_range = false;
  explicit uniform_01(base_type & rng);
  result_type operator()();
  result_type min() const;
  result_type max() const;
};
</pre>

<h3>Description</h3>

The distribution function <code>uniform_01</code> models a
<a href="random-concepts.html#random-dist">random distribution</a>.
On each invocation, it returns a random floating-point value uniformly
distributed in the range [0..1).

The value is computed using
<code>std::numeric_limits&lt;RealType&gt;::digits</code> random binary
digits, i.e. the mantissa of the floating-point value is completely
filled with random bits. [<em>Note:</em> Should this be configurable?]

<p>
The template parameter <code>RealType</code> shall denote a float-like
value type with support for binary operators +, -, and /.  It must be
large enough to hold floating-point numbers of value
<code>rng.max()-rng.min()+1</code>.
<p>
<code>base_type::result_type</code> must be a number-like value type,
it must support <code>static_cast&lt;&gt;</code> to
<code>RealType</code> and binary operator -.

<p>

<em>Note:</em> The current implementation is buggy, because it may not
fill all of the mantissa with random bits.  I'm unsure how to fill a
(to-be-invented) <code>boost::bigfloat</code> class with random bits
efficiently.  It's probably time for a traits class.

<h3>Members</h3>

<pre>explicit uniform_01(base_type & rng)</pre>

<strong>Effects:</strong> Constructs a <code>uniform_01</code> functor
with the given uniform random number generator as the underlying
source of random numbers.


<h2><a name="uniform_real">Class template <code>uniform_real</code></a></h2>

<h3>Synopsis</h3>

<pre>
#include &lt;<a href="../../boost/random/uniform_real.hpp">boost/random/uniform_real.hpp</a>&gt;

template&lt;class RealType = double&gt;
class uniform_real
{
public:
  typedef RealType input_type;
  typedef RealType result_type;
  static const bool has_fixed_range = false;
  uniform_real(RealType min = RealType(0), RealType max = RealType(1));
  result_type min() const;
  result_type max() const;
  void reset();
  template&lt;class UniformRandomNumberGenerator&gt;
  result_type operator()(UniformRandomNumberGenerator& urng);
};
</pre>

<h3>Description</h3>

The distribution function <code>uniform_real</code> models a
<a href="random-concepts.html#random-dist">random distribution</a>.
On each invocation, it returns a random floating-point
value uniformly distributed in the range [min..max). The value is
computed using
<code>std::numeric_limits&lt;RealType&gt;::digits</code> random binary
digits, i.e. the mantissa of the floating-point value is completely
filled with random bits.
<p>

<em>Note:</em> The current implementation is buggy, because it may not
fill all of the mantissa with random bits.


<h3>Members</h3>

<pre>    uniform_real(RealType min = RealType(0), RealType max = RealType(1))</pre>
<strong>Requires:</strong> min &lt;= max
<br>
<strong>Effects:</strong> Constructs a
<code>uniform_real</code> object; <code>min</code> and
<code>max</code> are the parameters of the distribution.

<pre>    result_type min() const</pre>
<strong>Returns:</strong> The "min" parameter of the distribution.

<pre>    result_type max() const</pre>
<strong>Returns:</strong> The "max" parameter of the distribution.


<h2><a name="bernoulli_distribution">Class template
<code>bernoulli_distribution</code></a></h2>

<h3>Synopsis</h3>

<pre>
#include &lt;<a href="../../boost/random/bernoulli_distribution.hpp">boost/random/bernoulli_distribution.hpp</a>&gt;

template&lt;class RealType = double>
class bernoulli_distribution
{
public:
  typedef int input_type;
  typedef bool result_type;

  explicit bernoulli_distribution(const RealType& p = RealType(0.5));
  RealType p() const;
  void reset();
  template&lt;class UniformRandomNumberGenerator&gt;
  result_type operator()(UniformRandomNumberGenerator& urng);
};
</pre>

<h3>Description</h3>

Instantiations of class template <code>bernoulli_distribution</code>
model a <a href="random-concepts.html#random-dist">random
distribution</a>.  Such a random distribution produces
<code>bool</code> values distributed with probabilities P(true) = p
and P(false) = 1-p.  p is the parameter of the distribution.

<h3>Members</h3>

<pre>    bernoulli_distribution(const RealType& p = RealType(0.5))</pre>

<strong>Requires:</strong> 0 &lt;= p &lt;= 1
<br>
<strong>Effects:</strong> Constructs a
<code>bernoulli_distribution</code> object.  <code>p</code> is the
parameter of the distribution.

<pre>    RealType p() const</pre>
<strong>Returns:</strong> The "p" parameter of the distribution.


<h2><a name="geometric_distribution">Class template
<code>geometric_distribution</code></a></h2>

<h3>Synopsis</h3>
<pre>
#include &lt;<a href="../../boost/random/geometric_distribution.hpp">boost/random/geometric_distribution.hpp</a>&gt;

template&lt;class UniformRandomNumberGenerator, class IntType = int&gt;
class geometric_distribution
{
public:
  typedef RealType input_type;
  typedef IntType result_type;

  explicit geometric_distribution(const RealType& p = RealType(0.5));
  RealType p() const;
  void reset();
  template&lt;class UniformRandomNumberGenerator&gt;
  result_type operator()(UniformRandomNumberGenerator& urng);
};
</pre>


<h3>Description</h3>

Instantiations of class template <code>geometric_distribution</code>
model a
<a href="random-concepts.html#random-dist">random distribution</a>.
A <code>geometric_distribution</code> random distribution produces
integer values <em>i</em> &gt;= 1 with p(i) = (1-p) * p<sup>i-1</sup>.
p is the parameter of the distribution.


<h3>Members</h3>

<pre>    geometric_distribution(const RealType& p = RealType(0.5))</pre>

<strong>Requires:</strong> 0 &lt; p &lt; 1
<br>
<strong>Effects:</strong> Constructs a
<code>geometric_distribution</code> object; <code>p</code> is the
parameter of the distribution.

<pre>   RealType p() const</pre>
<strong>Returns:</strong> The "p" parameter of the distribution.


<h2><a name="triangle_distribution">Class template
<code>triangle_distribution</code></a></h2>

<h3>Synopsis</h3>
<pre>
#include &lt;<a href="../../boost/random/triangle_distribution.hpp">boost/random/triangle_distribution.hpp</a>&gt;

template&lt;class RealType = double&gt;
class triangle_distribution
{
public:
  typedef RealType input_type;
  typedef RealType result_type;
  triangle_distribution(result_type a, result_type b, result_type c);
  result_type a() const;
  result_type b() const;
  result_type c() const;
  void reset();
  template&lt;class UniformRandomNumberGenerator&gt;
  result_type operator()(UniformRandomNumberGenerator& urng);
};
</pre>

<h3>Description</h3>

Instantiations of class template <code>triangle_distribution</code>
model a <a href="random-concepts.html#random-dist">random
distribution</a>.  The returned floating-point values <code>x</code>
satisfy <code>a <= x <= c</code>; <code>x</code> has a triangle
distribution, where <code>b</code> is the most probable value for
<code>x</code>.

<h3>Members</h3>

<pre>triangle_distribution(result_type a, result_type b, result_type c)</pre>

<strong>Effects:</strong> Constructs a
<code>triangle_distribution</code> functor. <code>a, b, c</code> are
the parameters for the distribution.
<p>


<h2><a name="exponential_distribution">Class template
<code>exponential_distribution</code></a></h2>

<h3>Synopsis</h3>
<pre>
#include &lt;<a href="../../boost/random/exponential_distribution.hpp">boost/random/exponential_distribution.hpp</a>&gt;

template&lt;class RealType = double&gt;
class exponential_distribution
{
public:
  typedef RealType input_type;
  typedef RealType result_type;
  explicit exponential_distribution(const result_type& lambda);
  RealType lambda() const;
  void reset();
  template&lt;class UniformRandomNumberGenerator&gt;
  result_type operator()(UniformRandomNumberGenerator& urng);
};
</pre>

<h3>Description</h3>

Instantiations of class template <code>exponential_distribution</code>
model a <a href="random-concepts.html#random-dist">random
distribution</a>.  Such a distribution produces random numbers x &gt;
0 distributed with probability density function p(x) = lambda *
exp(-lambda * x), where lambda is the parameter of the distribution.

<h3>Members</h3>

<pre>    exponential_distribution(const result_type& lambda = result_type(1))</pre>
<strong>Requires:</strong> lambda &gt; 0
<br>
<strong>Effects:</strong> Constructs an
<code>exponential_distribution</code> object with <code>rng</code> as
the reference to the underlying source of random
numbers. <code>lambda</code> is the parameter for the distribution.

<pre>    RealType lambda() const</pre>
<strong>Returns:</strong> The "lambda" parameter of the distribution.


<h2><a name="normal_distribution">Class template
<code>normal_distribution</code></a></h2>

<h3>Synopsis</h3>

<pre>
#include &lt;<a href="../../boost/random/normal_distribution.hpp">boost/random/normal_distribution.hpp</a>&gt;

template&lt;class RealType = double&gt;
class normal_distribution
{
public:
  typedef RealType input_type;
  typedef RealType result_type;
  explicit normal_distribution(const result_type& mean = 0,
			       const result_type& sigma = 1);
  RealType mean() const;
  RealType sigma() const;
  void reset();
  template&lt;class UniformRandomNumberGenerator&gt;
  result_type operator()(UniformRandomNumberGenerator& urng);
};
</pre>

<h3>Description</h3>

Instantiations of class template <code>normal_distribution</code>
model a <a href="random-concepts.html#random-dist">random
distribution</a>.  Such a distribution produces random numbers x
distributed with probability density function p(x) =
1/sqrt(2*pi*sigma) * exp(- (x-mean)<sup>2</sup> /
(2*sigma<sup>2</sup>) ), where mean and sigma are the parameters of
the distribution.


<h3>Members</h3>

<pre>
    explicit normal_distribution(const result_type& mean = 0,
                                 const result_type& sigma = 1);
</pre>

<strong>Requires:</strong> sigma &gt; 0
<br>
<strong>Effects:</strong> Constructs a
<code>normal_distribution</code> object; <code>mean</code> and
<code>sigma</code> are the parameters for the distribution.

<pre>    RealType mean() const</pre>
<strong>Returns:</strong> The "mean" parameter of the distribution.

<pre>    RealType sigma() const</pre>
<strong>Returns:</strong> The "sigma" parameter of the distribution.


<h2><a name="lognormal_distribution">Class template
<code>lognormal_distribution</code></a></h2>

<h3>Synopsis</h3>

<pre>
#include &lt;<a href="../../boost/random/lognormal_distribution.hpp">boost/random/lognormal_distribution.hpp</a>&gt;

template&lt;class RealType = double&gt;
class lognormal_distribution
{
public:
  typedef typename normal_distribution&lt;RealType&gt;::input_type
  typedef RealType result_type;
  explicit lognormal_distribution(const result_type& mean = 1.0,
			          const result_type& sigma = 1.0);
  RealType& mean() const;
  RealType& sigma() const;                                 
  void reset();
  template&lt;class UniformRandomNumberGenerator&gt;
  result_type operator()(UniformRandomNumberGenerator& urng);
};
</pre>

<h3>Description</h3>

Instantiations of class template <code>lognormal_distribution</code>
model a <a href="random-concepts.html#random-dist">random
distribution</a>.  Such a distribution produces random numbers
with p(x) = 1/(x * normal_sigma * sqrt(2*pi)) * exp(
-(log(x)-normal_mean)<sup>2</sup> / (2*normal_sigma<sup>2</sup>) )
for x > 0,
where normal_mean = log(mean<sup>2</sup>/sqrt(sigma<sup>2</sup>
+ mean<sup>2</sup>))
and normal_sigma = sqrt(log(1 + sigma<sup>2</sup>/mean<sup>2</sup>)).


<h3>Members</h3>

<pre>lognormal_distribution(const result_type& mean,
	   	       const result_type& sigma)</pre>

<strong>Effects:</strong> Constructs a
<code>lognormal_distribution</code> functor. <code>mean</code> and
<code>sigma</code> are the mean and standard deviation of the
lognormal distribution.
<p>


<h2><a name="uniform_on_sphere">Class template
<code>uniform_on_sphere</code></a></h2>

<h3>Synopsis</h3>

<pre>
#include &lt;<a href="../../boost/random/uniform_on_sphere.hpp">boost/random/uniform_on_sphere.hpp</a>&gt;

template&lt;class RealType = double,
  class Cont = std::vector&lt;RealType&gt; &gt;
class uniform_on_sphere
{
public:
  typedef RealType input_type;
  typedef Cont result_type;
  explicit uniform_on_sphere(int dim = 2);
  void reset();
  template&lt;class UniformRandomNumberGenerator&gt;
  const result_type & operator()(UniformRandomNumberGenerator& urng);
};
</pre>

<h3>Description</h3>

Instantiations of class template <code>uniform_on_sphere</code> model a
<a href="random-concepts.html#random-dist">random distribution</a>.
Such a distribution produces random numbers uniformly distributed on
the unit sphere of arbitrary dimension <code>dim</code>.  The
<code>Cont</code> template parameter must be a STL-like container type
with <code>begin</code> and <code>end</code> operations returning
non-const ForwardIterators of type <code>Cont::iterator</code>.

<h3>Members</h3>

<pre>explicit uniform_on_sphere(int dim = 2)</pre>

<strong>Effects:</strong> Constructs a <code>uniform_on_sphere</code>
functor. <code>dim</code> is the dimension of the sphere.
<p>

<p>
<hr>
Jens Maurer, 2003-10-25

</body>
</html>