/**
* Probability density function
*
* xᵃ⁻¹(1 - x)ᵝ⁻¹
* pdf = --------------
* B(α,β)
*
* @param number $α shape parameter α > 0
* @param number $β shape parameter β > 0
* @param number $x x ∈ (0,1)
*
* @return float
*/
public static function PDF($x, $α, $β)
{
Support::checkLimits(self::LIMITS, ['x' => $x, 'α' => $α, 'β' => $β]);
$xᵃ⁻¹ = pow($x, $α - 1);
$⟮1 − x⟯ᵝ⁻¹ = pow(1 - $x, $β - 1);
$B⟮α、β⟯ = Special::beta($α, $β);
return $xᵃ⁻¹ * $⟮1 − x⟯ᵝ⁻¹ / $B⟮α、β⟯;
}
/**
* Probability density function
*
* __________________
* / (d₁ x)ᵈ¹ d₂ᵈ²
* / ----------------
* √ (d₁ x + d₂)ᵈ¹⁺ᵈ²
* ---------------------
* / d₁ d₂ \
* x B | --, -- |
* \ 2 2 /
*
* @param number $x percentile ≥ 0
* @param int $d₁ degree of freedom v1 > 0
* @param int $d₂ degree of freedom v2 > 0
*
* @todo how to handle x = 0
*
* @return number probability
*/
public static function PDF($x, int $d₁, int $d₂)
{
Support::checkLimits(self::LIMITS, ['x' => $x, 'd₁' => $d₁, 'd₂' => $d₂]);
// Numerator
$⟮d₁x⟯ᵈ¹d₂ᵈ² = ($d₁ * $x) ** $d₁ * $d₂ ** $d₂;
$⟮d₁x＋d₂⟯ᵈ¹⁺ᵈ² = ($d₁ * $x + $d₂) ** ($d₁ + $d₂);
$√⟮d₁x⟯ᵈ¹d₂ᵈ²／⟮d₁x＋d₂⟯ᵈ¹⁺ᵈ² = sqrt($⟮d₁x⟯ᵈ¹d₂ᵈ² / $⟮d₁x＋d₂⟯ᵈ¹⁺ᵈ²);
// Denominator
$xB⟮d₁／2、d₂／2⟯ = $x * Special::beta($d₁ / 2, $d₂ / 2);
return $√⟮d₁x⟯ᵈ¹d₂ᵈ²／⟮d₁x＋d₂⟯ᵈ¹⁺ᵈ² / $xB⟮d₁／2、d₂／2⟯;
}