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<!DOCTYPE html>
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<meta name="description" content="StatsLib is a templated C++ library for fast computation of statistical distribution functions.">
<meta name="author" content="Keith O'Hara">
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<a data-toggle="collapse" href="#collapse1"><h4><strong style="font-size: 120%;">StatsLib: Beta Distribution</strong></h4></a>
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<div class="panel-body">
<a href="#intro">Introduction</a> <br>
<a href="#density">Density</a> <br>
<a href="#probability">Probability</a> <br>
<a href="#quantile">Quantile</a> <br>
<a href="#random">Random Sampling</a>
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<h3 style="text-align: left;" id="intro"><strong style="font-size: 120%;">Introduction</strong></h3>
<p>The density function of the Beta distribution:</p>
$$
f(x; \alpha, \beta) = \frac{1}{\mathcal{B}(\alpha,\beta)} x^{\alpha - 1} (1-x)^{\beta-1}
$$
<p>where $\mathcal{B}(\alpha,\beta)$ is the beta function.</p>
<p>Parameters:</p>
<ul>
<li><code>a_par</code> is $\alpha$</li>
<li><code>b_par</code> is $\beta$</li>
</ul>
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<h3 style="text-align: left;" id="density"><strong style="font-size: 120%;">Density</strong></h3>
<br>
<p><strong>Definition:</strong></p>
<pre class="brush: cpp;">
template<typename Ta, typename Tb>
statslib_constexpr
return_t<Ta> dbeta(const Ta x, const Tb a_par, const Tb b_par, const bool log_form = false);
</pre>
<p>Computes the density function.</p>
<hr>
<p><strong>Examples:</strong></p>
<pre class="brush: cpp;">
// parameters
double alpha = 3.0;
double beta = 2.0;
// standard input
double dens_val = stats::dbeta(0.5,alpha,beta);
double log_dens_val = stats::dbeta(0.5,alpha,beta,true);
// Armadillo input
arma::mat X(10,1);
X.fill(0.5);
arma::mat dens_vals_mat = stats::dbeta(X,alpha,beta);
arma::mat log_dens_vals_mat = stats::dbeta(X,alpha,beta,true);
</pre>
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<h3 style="text-align: left;" id="probability"><strong style="font-size: 120%;">Probability</strong></h3>
<br>
<p><strong>Definition:</strong></p>
<pre class="brush: cpp;">
template<typename Ta, typename Tb>
statslib_constexpr
return_t<Ta> pbeta(const Ta x, const Tb a_par, const Tb b_par, const bool log_form = false);
</pre>
<p>Computes the cumulative distribution function (CDF).</p>
<hr>
<p><strong>Examples:</strong></p>
<pre class="brush: cpp;">
// parameters
double alpha = 3.0;
double beta = 2.0;
// standard input
double prob_val = stats::pbeta(0.5,alpha,beta);
double log_prob_val = stats::pbeta(0.5,alpha,beta,true);
// Armadillo input
arma::mat X(10,1);
X.fill(0.5);
arma::mat prob_vals_mat = stats::pbeta(X,alpha,beta);
arma::mat log_prob_vals_mat = stats::pbeta(X,alpha,beta,true);
</pre>
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<h3 style="text-align: left;" id="quantile"><strong style="font-size: 120%;">Quantile</strong></h3>
<br>
<p><strong>Definition:</strong></p>
<pre class="brush: cpp;">
template<typename Ta, typename Tb>
statslib_constexpr
Ta qbeta(const Ta p, const Tb a_par, const Tb b_par);
</pre>
<p>Computes the quantile function.</p>
<hr>
<p><strong>Examples:</strong></p>
<pre class="brush: cpp;">
// parameters
double alpha = 3.0;
double beta = 2.0;
// standard input
double quant_val = stats::qbeta(0.7,alpha,beta);
// Armadillo input
arma::mat X(10,1);
X.fill(0.7);
arma::mat quant_vals_mat = stats::qbeta(X,alpha,beta);
</pre>
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<h3 style="text-align: left;" id="random"><strong style="font-size: 120%;">Random Sampling</strong></h3>
<br>
<p><strong>Definition:</strong></p>
<pre class="brush: cpp;">
// random engine seeding
template<typename T>
statslib_inline
return_t<T> rbeta(const T a_par, const T b_par, rand_engine_t& engine);
// seeding values
template<typename T>
statslib_inline
return_t<T> rbeta(const T a_par, const T b_par, uint_t seed_val = std::random_device{}());
// matrix output
template<typename mT, typename eT>
statslib_inline
mT rbeta(const uint_t n, const uint_t k, const eT a_par, const eT b_par);
</pre>
<p>Generates pseudo-random draws.</p>
<hr>
<p><strong>Examples:</strong></p>
<pre class="brush: cpp;">
// parameters
double alpha = 3.0;
double beta = 2.0;
// standard input
double rand_val = stats::rbeta(alpha,beta);
// Armadillo output: 10 x 1 matrix
arma::mat rand_mat = stats::rbeta<arma::mat>(10,1,alpha,beta);
</pre>
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