Overview

This is the API documentation for all statistical distribution procedures. There is a probability density function (PDF), cumulative distribution function (CDF), and quantile function or percent point function (PPF) for each distribution.

Normal Distribution

fsml_norm_pdf

Description

Probability density function for the normal distribution. $$f(x) = \frac{1}{\sigma \cdot \sqrt{2 \cdot \pi}} e^{ -\frac{1}{2} \cdot \left( \frac{x - \mu}{\sigma} \right)^2 }$$

The location parameter $\mu$ (mu) and scale parameter $\sigma$ (sigma) are optional arguments.

Syntax

result = fsml_norm_pdf(x [, mu, sigma])

Parameters

x: A scalar of type real.

mu: An optional argument and scalar of type real. It will default to 0.0 if not passed.

sigma: An optional argument and positive scalar of type real. If passed, it must be non-zero positive. It will default to 1.0 if not passed.

Invalid argument values will result in the return of a sentinel value.

Returns

The result is a scalar of the same type as x.

fsml_norm_cdf

Description

Cumulative distribution function $F(x) = \mathbb{P}(X \leq x)$ for normal distribution.

The location parameter $\mu$ (mu), scale parameter $\sigma$ (sigma), and tail option (tail) are optional arguments.

Syntax

result = fsml_norm_cdf(x [, mu, sigma, tail])

Parameters

x: A scalar of type real.

mu: An optional argument and scalar of type real. It will default to 0.0 if not passed.

sigma: An optional argument and positive scalar of type real. If passed, it must be non-zero positive. It will default to 1.0 if not passed.

tail: An optional argument and character string. If passed, it must be one of the following: "left", "right", "two", or "confidence". If not passed, it will default to "left".

Invalid argument values will result in the return of a sentinel value.

Returns

The result is a scalar of the same type as x.

fsml_norm_ppf

Description

Percent point function/quantile function $Q(p) = {F}_{x}^{-1}(p)$ for normal distribution.

It computes the position (x) based on the probability (p). The location parameter $\mu$ (mu) and scale parameter $\sigma$ (sigma) are optional arguments.

Note: The procedure uses bisection method. Conditions p=0.0 and p=1.0 cannot return negative and positive infinity; will return large negative or positive numbers (highly dependent on the tolerance threshold).

Syntax

result = fsml_norm_ppf(p [, mu, sigma])

Parameters

p: A scalar of type real. It must be between 0.0 and 1.0.

mu: An optional argument and scalar of type real. It will default to 0.0 if not passed.

sigma: An optional argument and positive scalar of type real. If passed, it must be non-zero positive. It will default to 1.0 if not passed.

Invalid argument values will result in the return of a sentinel value.

Returns

The result is a scalar of the same type as p.

Student's t Distribution

fsml_t_pdf

Description

Probability density function for student t distribution. Uses intrinsic gamma function (Fortran 2008 and later). $$f(x) = \frac{\Gamma\left(\frac{\nu + 1}{2}\right)}{\sqrt{\nu \cdot \pi}\, \cdot \Gamma\left(\frac{\nu}{2}\right)} \left(1 + \frac{x^2}{\nu}\right)^{-\frac{\nu + 1}{2}}$$ where $v$ = degrees of freedom (df) and $\Gamma$ is the gamma function.

The location parameter $\mu$ (mu) and scale parameter $\sigma$ (sigma) are optional arguments.

Syntax

result = fsml_t_pdf(x, df [, mu, sigma])

Parameters

x: A scalar of type real.

df: A scalar of type real. It must be 1.0 or higher.

mu: An optional argument and scalar of type real. It will default to 0.0 if not passed.

sigma: An optional argument and positive scalar of type real. If passed, it must be non-zero positive. It will default to 1.0 if not passed.

Invalid argument values will result in the return of a sentinel value.

Returns

The result is a scalar of the same type as x.

fsml_t_cdf

Description

Cumulative distribution function $F(x) = \mathbb{P}(X \leq x)$ for student t distribution.

The location parameter $\mu$ (mu), scale parameter $\sigma$ (sigma), and tail option (tail) are optional arguments.

Syntax

result = fsml_t_cdf(x, df [, mu, sigma, tail])

Parameters

x: A scalar of type real.

df: A scalar of type real. It must be 1.0 or higher.

mu: An optional argument and scalar of type real. It will default to 0.0 if not passed.

sigma: An optional argument and positive scalar of type real. If passed, it must be non-zero positive. It will default to 1.0 if not passed.

tail: An optional argument and character string. If passed, it must be one of the following: "left", "right", "two", or "confidence". If not passed, it will default to "left".

Invalid argument values will result in the return of a sentinel value.

Returns

The result is a scalar of the same type as x.

fsml_t_ppf

Description

Percent point function/quantile function $Q(p) = {F}_{x}^{-1}(p)$ for t distribution.

It computes the position (x) based on the probability (p) and degrees of freedom (df). The location parameter $\mu$ (mu) and scale parameter $\sigma$ (sigma) are optional arguments.

Note: The procedure uses bisection method. Conditions p=0.0 and p=1.0 cannot return negative and positive infinity; will return large negative or positive numbers (highly dependent on the tolerance threshold).

Syntax

result = fsml_t_ppf(p, df [, mu, sigma])

Parameters

p: A scalar of type real. It must be between 0.0 and 1.0.

df: A scalar of type real. It must be 1.0 or higher.

mu: An optional argument and scalar of type real. It will default to 0.0 if not passed.

sigma: An optional argument and positive scalar of type real. If passed, it must be non-zero positive. It will default to 1.0 if not passed.

Invalid argument values will result in the return of a sentinel value.

Returns

The result is a scalar of the same type as p.

Gamma Distribution

fsml_gamma_pdf

Description

Probability density function for gamma distribution. Uses intrinsic exp function. $$f(x) = \frac{\lambda^\alpha}{\Gamma(\alpha)} \cdot x^{\alpha - 1} \cdot e^{-\lambda \cdot x}, \quad x > 0, \ \alpha > 0, \ \lambda > 0$$

The equation can also be expressed with the scale parameter $\beta$, which is the inverse of the rate parameter $\lambda$, so that $\beta = \frac{1}{\lambda}$.

$$f(x) = \frac{1}{\Gamma(\alpha) \, \beta^\alpha} \cdot x^{\alpha - 1} \cdot e^{-x/\beta}, \quad x > 0,\ \alpha > 0,\ \beta > 0$$

The scale parameters $\alpha$ (alpha) and $\beta$ (beta) and the location parameter (loc) are optional arguments. The location parameter will shift the distribution in the manner that $\mu$ does for the normal distribution.

Syntax

result = fsml_gamma_pdf(x [, alpha, beta, loc])

Parameters

x: A scalar of type real.

alpha: An optional argument and positive scalar of type real. If passed, it must be non-zero positive. It will default to 1.0 if not passed.

beta: An optional argument and positive scalar of type real. If passed, it must be non-zero positive. It will default to 1.0 if not passed.

loc: A scalar of type real.

Invalid argument values will result in the return of a sentinel value.

Returns

The result is a scalar of the same type as x.

fsml_gamma_cdf

Description

Cumulative distribution function $F(x) = \mathbb{P}(X \leq x)$ for gamma distribution.

The scale parameters $\alpha$ (alpha) and $\beta$ (beta), the location parameter (loc), and tail option (tail) are optional arguments. The location parameter will shift the distribution in the manner that $\mu$ does for the normal distribution.

Syntax

result = fsml_gamma_cdf(x [, alpha, beta, loc, tail])

Parameters

x: A scalar of type real.

alpha: An optional argument and positive scalar of type real. If passed, it must be non-zero positive. It will default to 1.0 if not passed.

beta: An optional argument and positive scalar of type real. If passed, it must be non-zero positive. It will default to 1.0 if not passed.

loc: A scalar of type real.

tail: An optional argument and character string. If passed, it must be one of the following: "left", "right", "two", or "confidence". If not passed, it will default to "left".

Invalid argument values will result in the return of a sentinel value.

Returns

The result is a scalar of the same type as x.

fsml_gamma_ppf

Description

Percent point function/quantile function $Q(p) = {F}_{x}^{-1}(p)$ for gamma distribution.

It computes the position (x) based on the probability (p). The scale parameters $\alpha$ (alpha) and $\beta$ (beta) and the location parameter (loc) are optional arguments. The location parameter will shift the distribution in the manner that $\mu$ does for the normal distribution.

Note: The procedure uses bisection method.

Syntax

result = fsml_gamma_ppf(p [, alpha, beta, loc])

Parameters

p: A scalar of type real. It must be between 0.0 and 1.0.

alpha: An optional argument and positive scalar of type real. If passed, it must be non-zero positive. It will default to 1.0 if not passed.

beta: An optional argument and positive scalar of type real. If passed, it must be non-zero positive. It will default to 1.0 if not passed.

loc: A scalar of type real.

Invalid argument values will result in the return of a sentinel value.

Returns

The result is a scalar of the same type as p.

Exponential Distribution

fsml_exp_pdf

Description

Probability density function for exponential distribution. Uses intrinsic exp function. $$f(x) = \lambda \cdot e^{-\lambda \cdot x}, \quad x \geq 0, \ \lambda > 0$$

The rate parameter $\lambda$ (lambda) and location parameter (loc) are optional arguments. The location parameter will shift the distribution in the manner that $\mu$ does for the normal distribution.

Syntax

result = fsml_exp_pdf(x [, lambda, loc])

Parameters

x: A scalar of type real.

lambda: An optional argument and positive scalar of type real. If passed, it must be non-zero positive. It will default to 1.0 if not passed.

loc: A scalar of type real.

Invalid argument values will result in the return of a sentinel value.

Returns

The result is a scalar of the same type as x.

fsml_exp_cdf

Description

Cumulative distribution function $F(x) = \mathbb{P}(X \leq x)$ for exponential distribution.

The rate parameter $\lambda$ (lambda), location parameter (loc), and tail option (tail) are optional arguments. The location parameter will shift the distribution in the manner that $\mu$ does for the normal distribution.

Syntax

result = fsml_exp_cdf(x [, lambda, loc, tail])

Parameters

x: A scalar of type real.

lambda: An optional argument and positive scalar of type real. If passed, it must be non-zero positive. It will default to 1.0 if not passed.

loc: A scalar of type real.

tail: An optional argument and character string. If passed, it must be one of the following: "left", "right", "two", or "confidence". If not passed, it will default to "left".

Invalid argument values will result in the return of a sentinel value.

Returns

The result is a scalar of the same type as x.

fsml_exp_ppf

Description

Percent point function/quantile function $Q(p) = {F}_{x}^{-1}(p)$ for exponential distribution. Procedure uses bisection method. p should be between 0.0 and 1.0.

It computes the position (x) based on the probability (p). The rate parameter $\lambda$ (lambda) and location parameter (loc) are optional arguments. The location parameter will shift the distribution in the manner that $\mu$ does for the normal distribution.

Syntax

result = fsml_exp_ppf(p [, lambda, loc])

Parameters

p: A scalar of type real.

lambda: An optional argument and positive scalar of type real. If passed, it must be non-zero positive. It will default to 1.0 if not passed.

loc: A scalar of type real.

Invalid argument values will result in the return of a sentinel value.

Returns

The result is a scalar of the same type as p.

Chi-Squared Distribution

fsml_chi2_pdf

Description

Probability density function for the chi-squared distribution. Uses intrinsic exp and gamma function. $$f(x) = \frac{x^{\frac{k}{2} - 1} \cdot e^{ - \frac{x}{2} }}{2^{\frac{k}{2}} \cdot \Gamma\left(\frac{k}{2}\right)}, \quad x \geq 0, \ k > 0$$ where $k$ = degrees of freedom (df) and $\Gamma$ is the gamma function.

The procedure calculates the probability based on the provided parameters x and df. The location (loc) and scale parameter (scale) are optional arguments. The parameters loc and scale shift and scale the distribution, respectively, by a specified factor.

Syntax

result = fsml_chi2_pdf(x, df [, loc, scale])

Parameters

x: A scalar of type real.

df: A scalar of type real. It must be 1.0 or higher.

loc: An optional argument and scalar of type real. It will default to 0.0 if not passed.

scale: An optional argument and positive scalar of type real. If passed, it must be non-zero positive. It will default to 1.0 if not passed.

Invalid argument values will result in the return of a sentinel value.

Returns

The result is a scalar of the same type as x.

fsml_chi2_cdf

Description

Cumulative distribution function $F(x) = \mathbb{P}(X \leq x)$ for the chi-squared distribution.

The procedure calculates the probability based on the provided parameters x and df. The location (loc), scale parameter (scale), and tail option (tail) arguments are optional arguments. The parameters loc and scale shift and scale the distribution, respectively, by a specified factor.

Syntax

result = fsml_chi2_cdf(x, df [, loc, scale, tail])

Parameters

x: A scalar of type real.

df: A scalar of type real. It must be 1.0 or higher.

loc: An optional argument and scalar of type real. It will default to 0.0 if not passed.

scale: An optional argument and positive scalar of type real. If passed, it must be non-zero positive. It will default to 1.0 if not passed.

tail: An optional argument and character string. If passed, it must be one of the following: "left", "right", "two", or "confidence". If not passed, it will default to "left".

Invalid argument values will result in the return of a sentinel value.

Returns

The result is a scalar of the same type as x.

fsml_chi2_ppf

Description

Percent point function/quantile function $Q(p) = {F}_{x}^{-1}(p)$ for the chi-squared distribution. Uses the bisection method for numerical inversion of the CDF.

It computes the position (x) based on the probability (p) and degrees of freedom (df). The location (loc) and scale parameter (scale) are optional arguments. The parameters loc and scale shift and scale the distribution, respectively, by a specified factor.

Syntax

result = fsml_chi2_ppf(p, df [, loc, scale])

Parameters

p: A scalar of type real.

df: A scalar of type real. It must be 1.0 or higher.

loc: An optional argument and scalar of type real. It will default to 0.0 if not passed.

scale: An optional argument and positive scalar of type real. If passed, it must be non-zero positive. It will default to 1.0 if not passed.

Invalid argument values will result in the return of a sentinel value.

Returns

The result is a scalar of the same type as p.

F Distribution

fsml_f_pdf

Description

Probability density function for the F distribution. $$f(x) = \frac{1}{\mathrm{B}\left(\frac{d_1}{2}, \frac{d_2}{2}\right)} \cdot \left( \frac{d_1}{d_2} \right)^{ \frac{d_1}{2} } \cdot \frac{x^{ \frac{d_1}{2} - 1}}{\left(1 + \frac{d_1}{d_2} x \right)^{ \frac{d_1 + d_2}{2} }}, \quad x > 0$$ where $d_1$ = numerator degrees of freedom, $d_2$ = denominator degrees of freedom and $B$ is the complete beta function. (Uses intrinsic gamma function for beta.)

The F distribution is the distribution of $X = \frac{U_1/d_1}{U_2/d_2}$, where $U_1$ and $U_2$ are are random variables with chi-square distributions with $d_1$ (d1) and $d_2$ (d1) degrees of freedom, respectively.

The procedure calculates the probability based on the provided parameters x, d1, and d2. The location (loc) and scale parameter (scale) are optional arguments. The parameters loc and scale shift and scale the distribution, respectively, by a specified factor.

Syntax

result = fsml_f_pdf(x, d1, d2 [, loc, scale])

Parameters

x: A scalar of type real.

d1: A scalar of type real. It must be 1.0 or higher.

d2: A scalar of type real. It must be 1.0 or higher.

loc: An optional argument and scalar of type real. It will default to 0.0 if not passed.

scale: An optional argument and positive scalar of type real. If passed, it must be non-zero positive. It will default to 1.0 if not passed.

Invalid argument values will result in the return of a sentinel value.

Returns

The result is a scalar of the same type as x.

fsml_f_cdf

Description

Cumulative density function $F(x) = \mathbb{P}(X \leq x)$ for the F distribution.

The procedure calculates the probability based on the provided parameters x, d1, and d2. The location (loc), scale parameter (scale), and tail option (tail) arguments are optional arguments. The parameters loc and scale shift and scale the distribution, respectively, by a specified factor.

Syntax

result = fsml_f_cdf(x, d1, d2 [, loc, scale, tail])

Parameters

x: A scalar of type real.

d1: A scalar of type real. It must be 1.0 or higher.

d2: A scalar of type real. It must be 1.0 or higher.

loc: An optional argument and scalar of type real. It will default to 0.0 if not passed.

scale: An optional argument and positive scalar of type real. If passed, it must be non-zero positive. It will default to 1.0 if not passed.

tail: An optional argument and character string. If passed, it must be one of the following: "left", "right", "two", or "confidence". If not passed, it will default to "left".

Invalid argument values will result in the return of a sentinel value.

Returns

The result is a scalar of the same type as x.

fsml_f_ppf

Description

Percent point function / quantile function $Q(p) = F^{-1}(p)$ for the F distribution. Uses the bisection method to numerically invert the CDF.

It computes the position (x) based on the probability (p) and degrees of freedom (d1 and d2). The location (loc) and scale parameter (scale) are optional arguments. The parameters loc and scale shift and scale the distribution, respectively, by a specified factor.

Syntax

result = fsml_f_ppf(p, d1, d2 [, loc, scale])

Parameters

p: A scalar of type real.

d1: A scalar of type real. It must be 1.0 or higher.

d2: A scalar of type real. It must be 1.0 or higher.

loc: An optional argument and scalar of type real. It will default to 0.0 if not passed.

scale: An optional argument and positive scalar of type real. If passed, it must be non-zero positive. It will default to 1.0 if not passed.

Invalid argument values will result in the return of a sentinel value.

Returns

The result is a scalar of the same type as p.

Generalised Pareto Distribution

fsml_gpd_pdf

Description

Probability density function for generalised pareto distribution. $$f(x) = \frac{1}{\sigma} \cdot \left ( 1 + \frac{\xi \cdot (x - \mu)}{\sigma} \right)^{-\frac{1}{\xi} - 1}, \quad x \geq \mu, \ \sigma > 0, \ \xi \in \mathbb{R}$$ where $\xi$(xi) is a shape parameter, $\mu$ (mu) is the location, and $\sigma$ (sigma) is the scale parameter.

The procedure calculates the probability based on the provided parameters x and xi. The location (mu) and scale parameter (sigma) are optional arguments.

Syntax

result = fsml_gpd_pdf(x, xi [, mu, sigma])

Parameters

x: A scalar of type real.

xi: A scalar of type real.

mu: An optional argument and scalar of type real. It will default to 0.0 if not passed.

sigma: An optional argument and positive scalar of type real. If passed, it must be non-zero positive. It will default to 1.0 if not passed.

Invalid argument values will result in the return of a sentinel value.

Returns

The result is a scalar of the same type as x.

fsml_gpd_cdf

Description

Cumulative distribution function $F(x) = \mathbb{P}(X \leq x)$ for generalised pareto distribution.

The procedure calculates the probability based on the provided parameters x and xi. The location (mu) and scale parameter (sigma), and tail option (tail) are optional arguments.

Syntax

result = fsml_gpd_cdf(x, xi [, mu, sigma, tail])

Parameters

x: A scalar of type real.

xi: A scalar of type real.

mu: An optional argument and scalar of type real. It will default to 0.0 if not passed.

sigma: An optional argument and positive scalar of type real. If passed, it must be non-zero positive. It will default to 1.0 if not passed.

tail: An optional argument and character string. If passed, it must be one of the following: "left", "right", "two", or "confidence". If not passed, it will default to "left".

Invalid argument values will result in the return of a sentinel value.

Returns

The result is a scalar of the same type as x.

fsml_gpd_ppf

Description

Percent point function/quantile function $Q(p) = {F}_{x}^{-1}(p)$ for generalised pareto distribution. Procedure uses bisection method. p must be between 0.0 and 1.0.

It computes the position (x) based on the probability (p) and xi. The location parameter $\mu$ (mu) and scale parameter $\sigma$ (sigma) are optional arguments.

Syntax

result = fsml_gpd_ppf(p, xi [, mu, sigma])

Parameters

p: A scalar of type real.

xi: A scalar of type real.

mu: An optional argument and scalar of type real. It will default to 0.0 if not passed.

sigma: An optional argument and positive scalar of type real. If passed, it must be non-zero positive. It will default to 1.0 if not passed.

Invalid argument values will result in the return of a sentinel value.

Returns

The result is a scalar of the same type as p.

Examples

program example_dst

! |--------------------------------------------------------------------|
! | fsml - fortran statistics and machine learning library             |
! |                                                                    |
! | about                                                              |
! | -----                                                              |
! | Examples for distributions functions (dst module)                  |
! |                                                                    |
! | license : MIT                                                      |
! | author  : Sebastian G. Mutz (sebastian@sebastianmutz.com)          |
! |--------------------------------------------------------------------|

  use :: fsml
  use :: fsml_ini ! import wp; alternatively: iso_fortran_env, wp => real64

  implicit none

  print*

  ! ---- Normal Distribution

  print*, "> normal distribution"

  ! standardised normal pdf with x=1
  print*, fsml_norm_pdf(1.0_wp, mu=0.0_wp, sigma=1.0_wp)
  ! 0.24197072451914337

  ! normal distribution pdf with x=0.5, mu=0.4 and sigma=1.2
  print*, fsml_norm_pdf(0.5_wp, mu=0.4_wp, sigma=1.2_wp)
  ! 0.33129955521528498

  ! normal distribution cdf with x=2.3, mu=0.0 and signma=1.0
  print*, fsml_norm_cdf(2.3_wp, mu=0.0_wp, sigma=1.0_wp, tail="left")
  ! 0.98927588997832416

  ! normal distribution ppf with p=0.3, mu=0.0 and signma=1.0
  print*, fsml_norm_ppf(0.3_wp, mu=0.0_wp, sigma=1.0_wp)
  ! -0.52440051270878030

  print*

  ! ---- Student T Distribution

  print*, "> student t distribution"

  ! standard t pdf with x=1.5 and df=10
  print*, fsml_t_pdf(1.5_wp, df=10.0_wp, mu=0.0_wp, sigma=1.0_wp)
  ! 0.12744479428709160

  ! t distribution pdf with x=0.5, df=200, ~mu=0.8 and ~sigma=1.2
  ! similar to norm due to high df
  print*, fsml_t_pdf(0.5_wp, df=200.0_wp, mu=0.4_wp, sigma=1.2_wp)
  ! 0.33087996676641318

  ! get confidence interval with t distribution cdf with x=2.3, df=10, ~mu=0.0 and ~signma=1.0
  print*, fsml_t_cdf(2.3_wp, df=10.0_wp, mu=0.0_wp, sigma=1.0_wp, tail="confidence")
  ! 0.95574568671571991

  ! t distribution ppf with p=0.9, df=15, ~mu=0.0 and ~signma=1.0
  print*, fsml_t_ppf(0.9_wp, df=15.0_wp, mu=0.0_wp, sigma=1.0_wp)
  ! 1.3406056078565598

  print*

  ! ---- Gamma Distribution

  print*, "> gamma distribution"

  ! standard gamma pdf with x=2.3, defaults for shape parameters (alpha=1.0, beta=1.0) and loc
  print*, fsml_gamma_pdf(2.3_wp, alpha=1.0_wp, beta=1.0_wp, loc=0.0_wp)
  ! 0.10025884372280375

  ! standard gamma pdf with x=2.3, defaults for shape parameters and location shift loc=0.5
  print*, fsml_gamma_pdf(2.3_wp, alpha=1.0_wp, beta=1.0_wp, loc=0.5_wp)
  ! 0.16529888822158656

  ! gamma cdf with x=1.2, alpha=1.2, beta=0.6 and no location shift, and right end of the tail (survival function)
  print*, fsml_gamma_cdf(1.2_wp, alpha=1.2_wp, beta=0.6_wp, loc=0.0_wp, tail="right")
  ! 0.18230123290900657

  ! gamma ppf for p=0.5, alpha=0.5, beta=1.1 and no location shift
  print*, fsml_gamma_ppf(0.5_wp, alpha=0.5_wp, beta=1.1_wp, loc=0.0_wp)
  ! 0.25021503271636902


  print*

  ! ---- Exponential Distribution

  print*, "> exponential distribution"

  ! standard exponential pdf with x=1.3 and defaults for lambda=1.0 and loc=0.0
  print*, fsml_exp_pdf(1.3_wp, lambda=1.0_wp, loc=0.0_wp)
  ! 0.27253179303401259

  ! exponential pdf with x=1.3 and defaults for lambda=0.2 and loc=1.0
  print*, fsml_exp_pdf(1.3_wp, lambda=0.2_wp, loc=1.0_wp)
  ! 0.18835290671684976

  ! exponential cdf with x=0.1 and defaults for lambda=1.0 and loc=0.0
  print*, fsml_exp_cdf(1.3_wp, lambda=1.0_wp, loc=0.0_wp)
  ! 0.72746820696598746

  ! exponential ppf with p=0.95 and defaults for lambda=1.0 and loc=0.0
  print*, fsml_exp_ppf(0.5_wp, lambda=1.0_wp, loc=0.0_wp)
  ! 0.69314718055931490

  print*

  ! ---- Chi-Squared Distribution

  print*, "> chi-squared distribution"

  ! chi squared pdf with x=20.0, df=20, loc=0.5 and scale=1.0
  print*, fsml_chi2_pdf(20.0_wp, df=20.0_wp, loc=0.5_wp, scale=1.0_wp)
  ! 6.3955413942221373E-002

  ! chi squared pdf with x=5.1, df=10, loc=0.0 and scale=0.5
  print*, fsml_chi2_pdf(5.1_wp, df=10.0_wp, loc=0.0_wp, scale=0.5_wp)
  ! 0.17185714984072062

  ! chi squared cdf with x=11.5, df=10, loc=0.0 and scale=1.0
  print*, fsml_chi2_cdf(11.5_wp, df=10.0_wp, loc=0.0_wp, scale=1.0_wp)
  ! 0.68008856946173257

  ! chi squared ppf with p=0.2, df=10, loc=2.0 and scale=1.2
  print*, fsml_chi2_ppf(0.2_wp, df=10.0_wp, loc=2.0_wp, scale=1.2_wp)
  ! 9.4148951072402269

  print*

  ! ---- F Distribution

  print*, "> F distribution"

  ! f distribution pdf with x=2.0, d1=5.0, d2=2.0, loc=0.5 and scale=1.0
  print*, fsml_f_pdf(2.0_wp, d1=5.0_wp, d2=2.0_wp, loc=0.5_wp, scale=1.0_wp)
  ! 0.19431184938882604

  ! f distribution pdf with x=1.1, d1=5.0, d2=10, loc=0.0 and scale=0.5
  print*, fsml_f_pdf(1.1_wp, d1=5.0_wp, d2=10.0_wp, loc=0.0_wp, scale=0.5_wp)
  ! 0.25925652075006661

  ! f distribution cdf with x=11.5, d1=20, d2=10, loc=0.0 and scale=1.0
  print*, fsml_f_cdf(11.5_wp, d1=20.0_wp, d2=10.0_wp, loc=0.0_wp, scale=1.0_wp, tail="left")
  ! 0.99981682497307667

  ! f distribution ppf with p=0.2, d1=10, d2=20, loc=2.0 and scale=1.2
  print*, fsml_f_ppf(0.2_wp, d1=10.0_wp, d2=20.0_wp, loc=0.0_wp, scale=1.2_wp)
  ! 0.71332945788242341

  print*

  ! ---- Generalised Pareto Distribution

  print*, "> generalised pareto distribution"

  ! GPD pdf with x=1.1, xi=1.2, mu=0.0 and sigma=1.0
  print*, fsml_gpd_pdf(1.1_wp, xi=1.2_wp, mu=0.0_wp, sigma=1.0_wp)
  ! 0.21376603130006952

  ! GPD pdf with x=1.9, xi=0.2, mu=0.0 and sigma=1.2
  print*, fsml_gpd_pdf(1.9_wp, xi=0.2_wp, mu=0.0_wp, sigma=1.2_wp)
  ! 0.15994243683480086

  ! GPD cdf with x=2.1, xi=2.7, mu=1.2 and sigma=1.0 (standard left tail)
  print*, fsml_gpd_cdf(2.1_wp, xi=2.7_wp, mu=1.2_wp, sigma=1.0_wp, tail="left")
  ! 0.36650539816689109

  ! GPD ppf with p=0.2, xi=0.7, mu=0.0 and sigma=1.0 (standard left tail)
  print*, fsml_gpd_ppf(0.2_wp, xi=0.7_wp, mu=1.0_wp, sigma=1.0_wp)
  ! 1.2415150853975381

  print*


end program example_dst