for the [@http://en.wikipedia.org/wiki/Beta_distribution beta distribution]
defined on the interval \[0,1\] is given by:
-f(x;[alpha],[beta]) = x[super[alpha] - 1] (1 - x)[super[beta] -1] / B([alpha], [beta])
+[expression f(x;[alpha],[beta]) = x[super[alpha] - 1] (1 - x)[super[beta] -1] / B([alpha], [beta])]
-where B([alpha], [beta]) is the
+where [role serif_italic B([alpha], [beta])] is the
[@http://en.wikipedia.org/wiki/Beta_function beta function],
implemented in this library as __beta. Division by the beta function
ensures that the pdf is normalized to the range zero to unity.
The following graph illustrates examples of the pdf for various values
-of the shape parameters. Note the [alpha] = [beta] = 2 (blue line)
+of the shape parameters. Note the ['[alpha] = [beta] = 2] (blue line)
is dome-shaped, and might be approximated by a symmetrical triangular
distribution.
[graph beta_pdf]
-If [alpha] = [beta] = 1, then it is a __space
+If [alpha] = [beta] = 1, then it is a[emspace]
[@http://en.wikipedia.org/wiki/Uniform_distribution_%28continuous%29 uniform distribution],
equal to unity in the entire interval x = 0 to 1.
-If [alpha] __space and [beta] __space are < 1, then the pdf is U-shaped.
+If [alpha] and [beta] are < 1, then the pdf is U-shaped.
If [alpha] != [beta], then the shape is asymmetric
and could be approximated by a triangle
whose apex is away from the centre (where x = half).
Two pairs of parameter estimators are provided.
-One estimates either [alpha] __space or [beta] __space
+One estimates either [alpha] or [beta]
from presumed-known mean and variance.
-The other pair estimates either [alpha] __space or [beta] __space from
+The other pair estimates either [alpha] or [beta] from
the cdf and x.
-It is also possible to estimate [alpha] __space and [beta] __space from
+It is also possible to estimate [alpha] and [beta] from
'known' mode & quantile. For example, calculators are provided by the
[@http://www.ausvet.com.au/pprev/content.php?page=PPscript
Pooled Prevalence Calculator] and
RealType mean, // Expected value of mean.
RealType variance); // Expected value of variance.
-Returns the unique value of [alpha][space] that corresponds to a
+Returns the unique value of [alpha] that corresponds to a
beta distribution with mean /mean/ and variance /variance/.
static RealType find_beta(
RealType mean, // Expected value of mean.
RealType variance); // Expected value of variance.
-Returns the unique value of [beta][space] that corresponds to a
+Returns the unique value of [beta] that corresponds to a
beta distribution with mean /mean/ and variance /variance/.
static RealType find_alpha(
RealType x, // x.
RealType probability); // probability cdf
-Returns the value of [alpha][space] that gives:
+Returns the value of [alpha] that gives:
`cdf(beta_distribution<RealType>(alpha, beta), x) == probability`.
static RealType find_beta(
RealType x, // probability x.
RealType probability); // probability cdf.
-Returns the value of [beta][space] that gives:
+Returns the value of [beta] that gives:
`cdf(beta_distribution<RealType>(alpha, beta), x) == probability`.
[h4 Non-member Accessor Functions]
[h4 Related distributions]
-The beta distribution with both [alpha] __space and [beta] = 1 follows a
+The beta distribution with both [alpha] and [beta] = 1 follows a
[@http://en.wikipedia.org/wiki/Uniform_distribution_%28continuous%29 uniform distribution].
The [@http://en.wikipedia.org/wiki/Triangular_distribution triangular]
is used when less precise information is available.
The [@http://en.wikipedia.org/wiki/Binomial_distribution binomial distribution]
-is closely related when [alpha] __space and [beta] __space are integers.
+is closely related when [alpha] and [beta] are integers.
-With integer values of [alpha] __space and [beta] __space the distribution B(i, j) is
+With integer values of [alpha] and [beta] the distribution B(i, j) is
that of the j-th highest of a sample of i + j + 1 independent random variables
uniformly distributed between 0 and 1.
The cumulative probability from 0 to x is thus
[h4 Implementation]
-In the following table /a/ and /b/ are the parameters [alpha][space] and [beta],
+In the following table /a/ and /b/ are the parameters [alpha] and [beta],
/x/ is the random variable, /p/ is the probability and /q = 1-p/.
[table
[[Function][Implementation Notes]]
-[[pdf]
- [f(x;[alpha],[beta]) = x[super[alpha] - 1] (1 - x)[super[beta] -1] / B([alpha], [beta])
+[[pdf][[role serif_italic f(x;[alpha],[beta]) = x[super[alpha] - 1] (1 - x)[super[beta] -1] / B([alpha], [beta])]
Implemented using __ibeta_derivative(a, b, x).]]
-
[[cdf][Using the incomplete beta function __ibeta(a, b, x)]]
[[cdf complement][__ibetac(a, b, x)]]
[[quantile][Using the inverse incomplete beta function __ibeta_inv(a, b, p)]]
[[kurtosis excess][ [equation beta_dist_kurtosis] ]]
[[kurtosis][`kurtosis + 3`]]
[[parameter estimation][ ]]
-[[alpha
-
- from mean and variance][`mean * (( (mean * (1 - mean)) / variance)- 1)`]]
-[[beta
-
- from mean and variance][`(1 - mean) * (((mean * (1 - mean)) /variance)-1)`]]
+[[alpha (from mean and variance)][`mean * (( (mean * (1 - mean)) / variance)- 1)`]]
+[[beta (from mean and variance)][`(1 - mean) * (((mean * (1 - mean)) /variance)-1)`]]
[[The member functions `find_alpha` and `find_beta`
from cdf and probability x
__ibeta_inva, and __ibeta_invb respectively.]]
[[`find_alpha`][`ibeta_inva(beta, x, probability)`]]
[[`find_beta`][`ibeta_invb(alpha, x, probability)`]]
-]
+] [/table]
[h4 References]
[@http://mathworld.wolfram.com/BetaDistribution.html Wolfram MathWorld]
-[endsect][/section:beta_dist beta]
+[endsect] [/section:beta_dist beta]
[/ beta.qbk
Copyright 2006 John Maddock and Paul A. Bristow.
projects / platform / upstream / boost.git / blobdiff