# 対数正規分布 Logarithmic Normal Distribution
# 引数 変数 平均 標準偏差 ($X, $Mean, $StandardDeviation)
# 戻り値 対数正規分布 (@LogarithmicNormalDistribution)
sub LOGARITHMICNORMALDISTRIBUTION{
	my ($X, $Mean, $StandardDeviation) = @_;
	my @LogarithmicNormalDistribution = ();
	my $SimpsonsRule = 0;
	
	# 標準偏差の確認
	if(($X < 0) || ($StandardDeviation <= 0)){
		return 0;
	}
	
	# 計算
	my $PowerSD = $StandardDeviation * $StandardDeviation;
	my $E = exp($Mean + ($PowerSD / 2));
	my $Var = (exp($PowerSD) - 1) * exp((2 * $Mean) + $PowerSD);
	my $LnE = log($E) / log(exp(1));
	my $LnVE = log(1 + ($Var / ($E * $E))) / log(exp(1));
	my $LogNormalMean = $LnE - ($LnVE / 2);
	my $LogNormalStandardDeviation = sqrt($LnVE);
	
	# 確率密度関数 Frequency Function
	$LogarithmicNormalDistribution[0] = &FREQUENCYFUNCTION($X, $LogNormalMean, $LogNormalStandardDeviation);
	
	# シンプソンの公式 Simpson's Rule
	$SimpsonsRule = &SIMPSONSRULE($X, $LogNormalMean, $LogNormalStandardDeviation);
	
	# 下側累積確率 Lower Cumulative Distribution
	$LogarithmicNormalDistribution[1] = $SimpsonsRule;
	# 上側累積確率 Upper Cumulative Distribution
	$LogarithmicNormalDistribution[2] = 1 - $SimpsonsRule;
	
	return @LogarithmicNormalDistribution;
}
	
# 確率密度関数 Frequency Function
# 引数 変数 平均 標準偏差 ($X, $Mean, $StandardDeviation)
# 戻り値 確率密度関数 (@FrequencyFunction)
sub FREQUENCYFUNCTION{
	my ($X, $Mean, $StandardDeviation) = @_;
	my $FrequencyFunction = 0;
	my $Pi = atan2(1, 1) * 4;
	
	# 変数の確認
	if($X <= 0){
		return 0;
	}
	
	# 計算
	my $Ln = log($X) / log(exp(1));
	my $Power = ((($Ln - $Mean) / $StandardDeviation) * (($Ln - $Mean) / $StandardDeviation));
	
	# 対数正規分布 Frequency Function
	$FrequencyFunction = (1 / ($StandardDeviation * $X * sqrt(2 * $Pi))) * exp(-($Power / 2));
	
	return $FrequencyFunction;
}
	
# シンプソンの公式 Simpson's Rule
# 引数 変数 平均 標準偏差 ($X, $Mean, $StandardDeviation)
# 戻り値 シンプソンの公式 (@SimpsonsRule)
sub SIMPSONSRULE{
	my ($X, $Mean, $StandardDeviation) = @_;
	my $SimpsonsRule = 0;
	my $a = 0;
	my $b = $X;
	my $N = 10000;
	my $h = ($b - $a) / $N;
	my $Sum = 0;
	my $SumA = 0;
	my $SumB = 0;
	
	# 計算
	$SumA = &FREQUENCYFUNCTION($a, $Mean, $StandardDeviation);
	$SumB = &FREQUENCYFUNCTION($b, $Mean, $StandardDeviation);
	
	# 分割数が少ないと値が大きくなるにつれて
	# エクセルで計算した値と誤差がでる
	for(my $i = 1; $i < $N; $i++){
		my $t = $a + ($i * $h);
		my $num = ($i % 2 == 0 ? 2 : 4);
		my $tmp = &FREQUENCYFUNCTION($t, $Mean, $StandardDeviation);
	
		$Sum += $num * $tmp;
	}
	
	# シンプソンの公式 Simpson's Rule
	$SimpsonsRule = ($SumA + $Sum + $SumB) * ($h / 3);
	
	return $SimpsonsRule;
}