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```
```use distribution::{Continuous, Gamma, Univariate};
use rand::distributions::Distribution;
use rand::Rng;
use statistics::*;
use Result;

/// Implements the [Erlang](https://en.wikipedia.org/wiki/Erlang_distribution)
/// distribution
/// which is a special case of the
/// [Gamma](https://en.wikipedia.org/wiki/Gamma_distribution)
/// distribution
///
/// # Examples
///
/// ```
/// use statrs::distribution::{Erlang, Continuous};
/// use statrs::statistics::Mean;
/// use statrs::prec;
///
/// let n = Erlang::new(3, 1.0).unwrap();
/// assert_eq!(n.mean(), 3.0);
/// assert!(prec::almost_eq(n.pdf(2.0), 0.270670566473225383788, 1e-15));
/// ```
#[derive(Debug, Copy, Clone, PartialEq)]
pub struct Erlang {
g: Gamma,
}

impl Erlang {
/// Constructs a new erlang distribution with a shape (k)
/// of `shape` and a rate (λ) of `rate`
///
/// # Errors
///
/// Returns an error if `shape` or `rate` are `NaN`.
/// Also returns an error if `shape == 0` or `rate <= 0.0`
///
/// # Examples
///
/// ```
/// use statrs::distribution::Erlang;
///
/// let mut result = Erlang::new(3, 1.0);
/// assert!(result.is_ok());
///
/// result = Erlang::new(0, 0.0);
/// assert!(result.is_err());
/// ```
pub fn new(shape: u64, rate: f64) -> Result<Erlang> {
Gamma::new(shape as f64, rate).map(|g| Erlang { g: g })
}

/// Returns the shape (k) of the erlang distribution
///
/// # Examples
///
/// ```
/// use statrs::distribution::Erlang;
///
/// let n = Erlang::new(3, 1.0).unwrap();
/// assert_eq!(n.shape(), 3);
/// ```
pub fn shape(&self) -> u64 {
self.g.shape() as u64
}

/// Returns the rate (λ) of the erlang distribution
///
/// # Examples
///
/// ```
/// use statrs::distribution::Erlang;
///
/// let n = Erlang::new(3, 1.0).unwrap();
/// assert_eq!(n.rate(), 1.0);
/// ```
pub fn rate(&self) -> f64 {
self.g.rate()
}
}

impl Distribution<f64> for Erlang {
fn sample<R: Rng + ?Sized>(&self, r: &mut R) -> f64 {
Distribution::sample(&self.g, r)
}
}

impl Univariate<f64, f64> for Erlang {
/// Calculates the cumulative distribution function for the erlang
/// distribution
/// at `x`
///
/// # Formula
///
/// ```ignore
/// γ(k, λx)  (k - 1)!
/// ```
///
/// where `k` is the shape, `λ` is the rate, and `γ` is the lower
/// incomplete gamma function
fn cdf(&self, x: f64) -> f64 {
self.g.cdf(x)
}
}

impl Min<f64> for Erlang {
/// Returns the minimum value in the domain of the
/// erlang distribution representable by a double precision
/// float
///
/// # Formula
///
/// ```ignore
/// 0
/// ```
fn min(&self) -> f64 {
self.g.min()
}
}

impl Max<f64> for Erlang {
/// Returns the maximum value in the domain of the
/// erlang distribution representable by a double precision
/// float
///
/// # Formula
///
/// ```ignore
/// INF
/// ```
fn max(&self) -> f64 {
self.g.max()
}
}

impl Mean<f64> for Erlang {
/// Returns the mean of the erlang distribution
///
/// # Remarks
///
/// Returns `shape` if `rate == f64::INFINITY`. This behavior
/// is borrowed from the Math.NET implementation
///
/// # Formula
///
/// ```ignore
/// k / λ
/// ```
///
/// where `k` is the shape and `λ` is the rate
fn mean(&self) -> f64 {
self.g.mean()
}
}

impl Variance<f64> for Erlang {
/// Returns the variance of the erlang distribution
///
/// # Formula
///
/// ```ignore
/// k / λ^2
/// ```
///
/// where `α` is the shape and `λ` is the rate
fn variance(&self) -> f64 {
self.g.variance()
}

/// Returns the standard deviation of the erlang distribution
///
/// # Formula
///
/// ```ignore
/// sqrt(k) / λ
/// ```
///
/// where `k` is the shape and `λ` is the rate
fn std_dev(&self) -> f64 {
self.g.std_dev()
}
}

impl Entropy<f64> for Erlang {
/// Returns the entropy of the erlang distribution
///
/// # Formula
///
/// ```ignore
/// k - ln(λ) + ln(Γ(k)) + (1 - k) * ψ(k)
/// ```
///
/// where `k` is the shape, `λ` is the rate, `Γ` is the gamma function,
/// and `ψ` is the digamma function
fn entropy(&self) -> f64 {
self.g.entropy()
}
}

impl Skewness<f64> for Erlang {
/// Returns the skewness of the erlang distribution
///
/// # Formula
///
/// ```ignore
/// 2 / sqrt(k)
/// ```
///
/// where `k` is the shape
fn skewness(&self) -> f64 {
self.g.skewness()
}
}

impl Mode<f64> for Erlang {
/// Returns the mode for the erlang distribution
///
/// # Remarks
///
/// Returns `shape` if `rate ==f64::INFINITY`. This behavior
/// is borrowed from the Math.NET implementation
///
/// # Formula
///
/// ```ignore
/// (k - 1) / λ
/// ```
///
/// where `k` is the shape and `λ` is the rate
fn mode(&self) -> f64 {
self.g.mode()
}
}

impl Continuous<f64, f64> for Erlang {
/// Calculates the probability density function for the erlang distribution
/// at `x`
///
/// # Remarks
///
/// Returns `NAN` if any of `shape` or `rate` are `INF`
/// or if `x` is `INF`
///
/// # Formula
///
/// ```ignore
/// (λ^k / Γ(k)) * x^(k - 1) * e^(-λ * x)
/// ```
///
/// where `k` is the shape, `λ` is the rate, and `Γ` is the gamma function
fn pdf(&self, x: f64) -> f64 {
self.g.pdf(x)
}

/// Calculates the log probability density function for the erlang
/// distribution
/// at `x`
///
/// # Remarks
///
/// Returns `NAN` if any of `shape` or `rate` are `INF`
/// or if `x` is `INF`
///
/// # Formula
///
/// ```ignore
/// ln((λ^k / Γ(k)) * x^(k - 1) * e ^(-λ * x))
/// ```
///
/// where `k` is the shape, `λ` is the rate, and `Γ` is the gamma function
fn ln_pdf(&self, x: f64) -> f64 {
self.g.ln_pdf(x)
}
}

#[cfg_attr(rustfmt, rustfmt_skip)]
#[cfg(test)]
mod test {
use std::f64;
use distribution::Erlang;
use distribution::internal::*;

fn try_create(shape: u64, rate: f64) -> Erlang {
let n = Erlang::new(shape, rate);
assert!(n.is_ok());
n.unwrap()
}

fn create_case(shape: u64, rate: f64) {
let n = try_create(shape, rate);
assert_eq!(shape, n.shape());
assert_eq!(rate, n.rate());
}

fn bad_create_case(shape: u64, rate: f64) {
let n = Erlang::new(shape, rate);
assert!(n.is_err());
}

#[test]
fn test_create() {
create_case(1, 0.1);
create_case(1, 1.0);
create_case(10, 10.0);
create_case(10, 1.0);
create_case(10, f64::INFINITY);
}

#[test]