public final class LogNormalDistribution extends Object
\( X \) is log-normally distributed if its natural logarithm \( \ln(x) \) is normally distributed. The probability density function of \( X \) is:
\[ f(x; \mu, \sigma) = \frac 1 {x\sigma\sqrt{2\pi\,}} e^{-{\frac 1 2}\left( \frac{\ln x-\mu}{\sigma} \right)^2 } \]
for \( \mu \) the mean of the normally distributed natural logarithm of this distribution, \( \sigma > 0 \) the standard deviation of the normally distributed natural logarithm of this distribution, and \( x \in (0, \infty) \).
ContinuousDistribution.Sampler| Modifier and Type | Method and Description |
|---|---|
ContinuousDistribution.Sampler |
createSampler(UniformRandomProvider rng)
Creates a sampler.
|
double |
cumulativeProbability(double x)
For a random variable
X whose values are distributed according
to this distribution, this method returns P(X <= x). |
double |
density(double x)
Returns the probability density function (PDF) of this distribution
evaluated at the specified point
x. |
double |
getMean()
Gets the mean of this distribution.
|
double |
getMu()
Gets the
mu parameter of this distribution. |
double |
getSigma()
Gets the
sigma parameter of this distribution. |
double |
getSupportLowerBound()
Gets the lower bound of the support.
|
double |
getSupportUpperBound()
Gets the upper bound of the support.
|
double |
getVariance()
Gets the variance of this distribution.
|
double |
inverseCumulativeProbability(double p)
Computes the quantile function of this distribution.
|
double |
inverseSurvivalProbability(double p)
Computes the inverse survival probability function of this distribution.
|
double |
logDensity(double x)
Returns the natural logarithm of the probability density function
(PDF) of this distribution evaluated at the specified point
x. |
static LogNormalDistribution |
of(double mu,
double sigma)
Creates a log-normal distribution.
|
double |
probability(double x0,
double x1)
For a random variable
X whose values are distributed according
to this distribution, this method returns P(x0 < X <= x1). |
double |
survivalProbability(double x)
For a random variable
X whose values are distributed according
to this distribution, this method returns P(X > x). |
public static LogNormalDistribution of(double mu, double sigma)
mu - Mean of the natural logarithm of the distribution values.sigma - Standard deviation of the natural logarithm of the distribution values.IllegalArgumentException - if sigma <= 0.public double getMu()
mu parameter of this distribution.
This is the mean of the natural logarithm of the distribution values,
not the mean of distribution.public double getSigma()
sigma parameter of this distribution.
This is the standard deviation of the natural logarithm of the distribution values,
not the standard deviation of distribution.public double density(double x)
x.
In general, the PDF is the derivative of the CDF.
If the derivative does not exist at x, then an appropriate
replacement should be returned, e.g. Double.POSITIVE_INFINITY,
Double.NaN, or the limit inferior or limit superior of the
difference quotient.
For mu, and sigma s of this distribution, the PDF
is given by
0 if x <= 0,exp(-0.5 * ((ln(x) - mu) / s)^2) / (s * sqrt(2 * pi) * x)
otherwise.x - Point at which the PDF is evaluated.x.public double probability(double x0, double x1)
X whose values are distributed according
to this distribution, this method returns P(x0 < X <= x1).
The default implementation uses the identity
P(x0 < X <= x1) = P(X <= x1) - P(X <= x0)probability in interface ContinuousDistributionx0 - Lower bound (exclusive).x1 - Upper bound (inclusive).x0 and x1, excluding the lower
and including the upper endpoint.public double logDensity(double x)
x.
See documentation of density(double) for computation details.
x - Point at which the PDF is evaluated.x.public double cumulativeProbability(double x)
X whose values are distributed according
to this distribution, this method returns P(X <= x).
In other words, this method represents the (cumulative) distribution
function (CDF) for this distribution.
For mu, and sigma s of this distribution, the CDF
is given by
0 if x <= 0,0 if ln(x) - mu < 0 and mu - ln(x) > 40 * s, as
in these cases the actual value is within Double.MIN_VALUE of 0,
1 if ln(x) - mu >= 0 and ln(x) - mu > 40 * s,
as in these cases the actual value is within Double.MIN_VALUE of
1,0.5 + 0.5 * erf((ln(x) - mu) / (s * sqrt(2)) otherwise.x - Point at which the CDF is evaluated.x.public double survivalProbability(double x)
X whose values are distributed according
to this distribution, this method returns P(X > x).
In other words, this method represents the complementary cumulative
distribution function.
By default, this is defined as 1 - cumulativeProbability(x), but
the specific implementation may be more accurate.
x - Point at which the survival function is evaluated.x.public double inverseCumulativeProbability(double p)
X distributed according to this distribution, the
returned value is:
\[ x = \begin{cases} \inf \{ x \in \mathbb R : P(X \le x) \ge p\} & \text{for } 0 \lt p \le 1 \\ \inf \{ x \in \mathbb R : P(X \le x) \gt 0 \} & \text{for } p = 0 \end{cases} \]
The default implementation returns:
ContinuousDistribution.getSupportLowerBound() for p = 0,ContinuousDistribution.getSupportUpperBound() for p = 1, orcumulativeProbability(x) - p.
The bounds may be bracketed for efficiency.inverseCumulativeProbability in interface ContinuousDistributionp - Cumulative probability.p-quantile of this distribution
(largest 0-quantile for p = 0).public double inverseSurvivalProbability(double p)
X distributed according to this distribution, the
returned value is:
\[ x = \begin{cases} \inf \{ x \in \mathbb R : P(X \ge x) \le p\} & \text{for } 0 \le p \lt 1 \\ \inf \{ x \in \mathbb R : P(X \ge x) \lt 1 \} & \text{for } p = 1 \end{cases} \]
By default, this is defined as inverseCumulativeProbability(1 - p), but
the specific implementation may be more accurate.
The default implementation returns:
ContinuousDistribution.getSupportLowerBound() for p = 1,ContinuousDistribution.getSupportUpperBound() for p = 0, orsurvivalProbability(x) - p.
The bounds may be bracketed for efficiency.inverseSurvivalProbability in interface ContinuousDistributionp - Survival probability.(1-p)-quantile of this distribution
(largest 0-quantile for p = 1).public double getMean()
For \( \mu \) the mean of the normally distributed natural logarithm of this distribution, \( \sigma > 0 \) the standard deviation of the normally distributed natural logarithm of this distribution, the mean is:
\[ \exp(\mu + \frac{\sigma^2}{2}) \]
public double getVariance()
For \( \mu \) the mean of the normally distributed natural logarithm of this distribution, \( \sigma > 0 \) the standard deviation of the normally distributed natural logarithm of this distribution, the variance is:
\[ [\exp(\sigma^2) - 1)] \exp(2 \mu + \sigma^2) \]
public double getSupportLowerBound()
inverseCumulativeProbability(0), i.e.
\( \inf \{ x \in \mathbb R : P(X \le x) \gt 0 \} \).
The lower bound of the support is always 0.
public double getSupportUpperBound()
inverseCumulativeProbability(1), i.e.
\( \inf \{ x \in \mathbb R : P(X \le x) = 1 \} \).
The upper bound of the support is always positive infinity.
positive infinity.public ContinuousDistribution.Sampler createSampler(UniformRandomProvider rng)
createSampler in interface ContinuousDistributionrng - Generator of uniformly distributed numbers.Copyright © 2018–2022 The Apache Software Foundation. All rights reserved.