/**
* Log normal distribution - cumulative distribution function
*
* https://en.wikipedia.org/wiki/Log-normal_distribution
*
* 1 1 / ln x - μ \
* cdf = - + - erf | -------- |
* 2 2 \ √2σ /
*
* @param number $x > 0
* @param number $μ location parameter
* @param number $σ scale parameter > 0
* @return number
*/
public static function CDF($x, $μ, $σ)
{
Support::checkLimits(self::LIMITS, ['x' => $x, 'μ' => $μ, 'σ' => $σ]);
$π = \M_PI;
$⟮ln x − μ⟯ = log($x) - $μ;
$√2σ = sqrt(2) * $σ;
return 1 / 2 + 1 / 2 * Special::erf($⟮ln x − μ⟯ / $√2σ);
}
/**
* Cumulative distribution function
* Probability of being below X.
* Area under the normal distribution from -∞ to X.
* _ _
* 1 | / x - μ \ |
* cdf(x) = - | 1 + erf| ----- | |
* 2 |_ \ σ√2 / _|
*
* @param number $x upper bound
* @param number $μ mean
* @param number $σ standard deviation
*
* @return float cdf(x) below
*/
public static function CDF($x, $μ, $σ) : float
{
Support::checkLimits(self::LIMITS, ['x' => $x, 'μ' => $μ, 'σ' => $σ]);
return 1 / 2 * (1 + Special::erf(($x - $μ) / ($σ * sqrt(2))));
}
