SQL functions for evaluating probability functions. More...

Functions
float8	bernoulli_cdf (float8 x, float8 sp)
	Bernoulli cumulative distribution function. More...

float8	bernoulli_pmf (int4 x, float8 sp)
	Bernoulli probability mass function. More...

float8	bernoulli_quantile (float8 p, float8 sp)
	Bernoulli quantile function. More...

float8	beta_cdf (float8 x, float8 alpha, float8 beta)
	Beta cumulative distribution function. More...

float8	beta_pdf (float8 x, float8 alpha, float8 beta)
	Beta probability density function. More...

float8	beta_quantile (float8 p, float8 alpha, float8 beta)
	Beta quantile function. More...

float8	binomial_cdf (float8 x, int4 n, float8 sp)
	Binomial cumulative distribution function. More...

float8	binomial_pmf (int4 x, int4 n, float8 sp)
	Binomial probability mass function. More...

float8	binomial_quantile (float8 p, int4 n, float8 sp)
	Binomial quantile function. More...

float8	cauchy_cdf (float8 x, float8 location, float8 scale)
	Cauchy cumulative distribution function. More...

float8	cauchy_pdf (float8 x, float8 location, float8 scale)
	Cauchy probability density function. More...

float8	cauchy_quantile (float8 p, float8 location, float8 scale)
	Cauchy quantile function. More...

float8	chi_squared_cdf (float8 x, float8 df)
	Chi-squared cumulative distribution function. More...

float8	chi_squared_pdf (float8 x, float8 df)
	Chi-squared distribution probability density function. More...

float8	chi_squared_quantile (float8 p, float8 df)
	Chi-squared distribution quantile function. More...

float8	exponential_cdf (float8 x, float8 lambda)
	Exponential cumulative distribution function. More...

float8	exponential_pdf (float8 x, float8 lambda)
	Exponential probability density function. More...

float8	exponential_quantile (float8 p, float8 lambda)
	Exponential quantile function. More...

float8	extreme_value_cdf (float8 x, float8 location, float8 scale)
	Extreme Value cumulative distribution function. More...

float8	extreme_value_pdf (float8 x, float8 location, float8 scale)
	Extreme Value probability density function. More...

float8	extreme_value_quantile (float8 p, float8 location, float8 scale)
	Extreme Value quantile function. More...

float8	fisher_f_cdf (float8 x, float8 df1, float8 df2)
	Fisher F cumulative distribution function. More...

float8	fisher_f_pdf (float8 x, float8 df1, float8 df2)
	Fisher F probability density function. More...

float8	fisher_f_quantile (float8 p, float8 df1, float8 df2)
	Fisher F quantile function. More...

float8	gamma_cdf (float8 x, float8 shape, float8 scale)
	Gamma cumulative distribution function. More...

float8	gamma_pdf (float8 x, float8 shape, float8 scale)
	Gamma probability density function. More...

float8	gamma_quantile (float8 p, float8 shape, float8 scale)
	Gamma quantile function. More...

float8	geometric_cdf (float8 x, float8 sp)
	Geometric cumulative distribution function. More...

float8	geometric_pmf (int4 x, float8 sp)
	Geometric probability mass function. More...

float8	geometric_quantile (float8 p, float8 sp)
	Geometric quantile function. More...

float8	hypergeometric_cdf (float8 x, int4 r, int4 n, int4 N)
	Hypergeometric cumulative distribution function. More...

float8	hypergeometric_pmf (int4 x, int4 r, int4 n, int4 N)
	Hypergeometric probability mass function. More...

float8	hypergeometric_quantile (float8 p, int4 r, int4 n, int4 N)
	Hypergeometric quantile function. More...

float8	inverse_gamma_cdf (float8 x, float8 shape, float8 scale)
	Inverse Gamma cumulative distribution function. More...

float8	inverse_gamma_pdf (float8 x, float8 shape, float8 scale)
	Inverse Gamma probability density function. More...

float8	inverse_gamma_quantile (float8 p, float8 shape, float8 scale)
	Inverse Gamma quantile function. More...

float8	kolmogorov_cdf (float8 x)
	Kolmogorov cumulative distribution function. More...

float8	laplace_cdf (float8 x, float8 mean, float8 scale)
	Laplace cumulative distribution function. More...

float8	laplace_pdf (float8 x, float8 mean, float8 scale)
	Laplace probability density function. More...

float8	laplace_quantile (float8 p, float8 mean, float8 scale)
	Laplace quantile function. More...

float8	logistic_cdf (float8 x, float8 mean, float8 scale)
	Logistic cumulative distribution function. More...

float8	logistic_pdf (float8 x, float8 mean, float8 scale)
	Logistic probability density function. More...

float8	logistic_quantile (float8 p, float8 mean, float8 scale)
	Logistic quantile function. More...

float8	lognormal_cdf (float8 x, float8 location, float8 scale)
	Log-normal cumulative distribution function. More...

float8	lognormal_pdf (float8 x, float8 location, float8 scale)
	Log-normal probability density function. More...

float8	lognormal_quantile (float8 p, float8 location, float8 scale)
	Log-normal quantile function. More...

float8	negative_binomial_cdf (float8 x, float8 r, float8 sp)
	Negative binomial cumulative distribution function. More...

float8	negative_binomial_pmf (int4 x, float8 r, float8 sp)
	Negative binomial probability mass function. More...

float8	negative_binomial_quantile (float8 p, float8 r, float8 sp)
	Negative binomial quantile function. More...

float8	non_central_beta_cdf (float8 x, float8 alpha, float8 beta, float8 ncp)
	Noncentral beta cumulative distribution function. More...

float8	non_central_beta_pdf (float8 x, float8 alpha, float8 beta, float8 ncp)
	Noncentral beta probability density function. More...

float8	non_central_beta_quantile (float8 p, float8 alpha, float8 beta, float8 ncp)
	Noncentral beta quantile function. More...

float8	non_central_chi_squared_cdf (float8 x, float8 df, float8 ncp)
	Noncentral chi-squared cumulative distribution function. More...

float8	non_central_chi_squared_pdf (float8 x, float8 df, float8 ncp)
	Noncentral chi-squared distribution probability density function. More...

float8	non_central_chi_squared_quantile (float8 p, float8 df, float8 ncp)
	Noncentral chi-squared distribution quantile function. More...

float8	non_central_f_cdf (float8 x, float8 df1, float8 df2, float8 ncp)
	Noncentral Fisher F cumulative distribution function. More...

float8	non_central_f_pdf (float8 x, float8 df1, float8 df2, float8 ncp)
	Noncentral Fisher F probability density function. More...

float8	non_central_f_quantile (float8 p, float8 df1, float8 df2, float8 ncp)
	Noncentral Fisher F quantile function. More...

float8	non_central_t_cdf (float8 x, float8 df, float8 ncp)
	Noncentral Student-t cumulative distribution function. More...

float8	non_central_t_pdf (float8 x, float8 df, float8 ncp)
	Noncentral Student-t probability density function. More...

float8	non_central_t_quantile (float8 p, float8 df, float8 ncp)
	Noncentral Student-t quantile function. More...

float8	normal_cdf (float8 x, float8 mean=0, float8 sd=1)
	Normal cumulative distribution function. More...

float8	normal_pdf (float8 x, float8 mean=0, float8 sd=1)
	Normal probability density function. More...

float8	normal_quantile (float8 p, float8 mean=0, float8 sd=1)
	Normal quantile function. More...

float8	pareto_cdf (float8 x, float8 scale, float8 shape)
	Pareto cumulative distribution function. More...

float8	pareto_pdf (float8 x, float8 scale, float8 shape)
	Pareto probability density function. More...

float8	pareto_quantile (float8 p, float8 scale, float8 shape)
	Pareto quantile function. More...

float8	poisson_cdf (float8 x, float8 mean)
	Poisson cumulative distribution function. More...

float8	poisson_pmf (int4 x, float8 mean)
	Poisson probability mass function. More...

float8	poisson_quantile (float8 p, float8 mean)
	Poisson quantile function. More...

float8	rayleigh_cdf (float8 x, float8 scale)
	Rayleigh cumulative distribution function. More...

float8	rayleigh_pdf (float8 x, float8 scale)
	Rayleigh probability density function. More...

float8	rayleigh_quantile (float8 p, float8 scale)
	Rayleigh quantile function. More...

float8	students_t_cdf (float8 x, float8 df)
	Student's t cumulative distribution function. More...

float8	students_t_pdf (float8 x, float8 df)
	Student's t probability density function. More...

float8	students_t_quantile (float8 p, float8 df)
	Student's t quantile function. More...

float8	triangular_cdf (float8 x, float8 lower, float8 mode, float8 upper)
	Triangular cumulative distribution function. More...

float8	triangular_pdf (float8 x, float8 lower, float8 mode, float8 upper)
	Triangular probability density function. More...

float8	triangular_quantile (float8 p, float8 lower, float8 mode, float8 upper)
	Triangular quantile function. More...

float8	uniform_cdf (float8 x, float8 lower, float8 upper)
	Uniform cumulative distribution function. More...

float8	uniform_pdf (float8 x, float8 lower, float8 upper)
	Uniform probability density function. More...

float8	uniform_quantile (float8 p, float8 lower, float8 upper)
	Uniform quantile function. More...

float8	weibull_cdf (float8 x, float8 shape, float8 scale)
	Weibull cumulative distribution function. More...

float8	weibull_pdf (float8 x, float8 shape, float8 scale)
	Weibull probability density function. More...

float8	weibull_quantile (float8 p, float8 shape, float8 scale)
	Weibull quantile function. More...

Detailed Description

See Also: For an overview of probability functions, see the module description Probability Functions.

Definition in file prob.sql_in.

Function Documentation

float8 bernoulli_cdf	(	float8	x,
		float8	sp
	)

Parameters

x	Random variate \( x \)
sp	Success probability \( p \in [0,1] \)

Returns: \( \Pr[X \leq x] \) where \( X \) is a Bernoulli-distributed random variable with success probability \( \mathit{sp} \)

Definition at line 138 of file prob.sql_in.

float8 bernoulli_pmf	(	int4	x,
		float8	sp
	)

Parameters

x	Random variate \( x \)
sp	Success probability \( \mathit{sp} \in [0,1] \)

Returns: \( f(x) \) where \( f \) is the probability mass function of a Bernoulli-distributed random variable with success probability \( \mathit{sp} \)

Definition at line 155 of file prob.sql_in.

float8 bernoulli_quantile	(	float8	p,
		float8	sp
	)

Parameters

p	Probability \( p \in [0,1] \)
sp	Success probability \( \mathit{sp} \in [0,1] \)

Returns: 0 if \( p \leq 1 - \mathit{sp} \) and 1 otherwise

Definition at line 170 of file prob.sql_in.

float8 beta_cdf	(	float8	x,
		float8	alpha,
		float8	beta
	)

Parameters

x	Random variate \( x \)
alpha	Shape \( \alpha > 0 \)
beta	Shape \( \beta > 0 \)

Returns: \( \Pr[X \leq x] \) where \( X \) is a beta distributed random variable with shape parameters \( \alpha \) and \( \beta \)

Definition at line 188 of file prob.sql_in.

float8 beta_pdf	(	float8	x,
		float8	alpha,
		float8	beta
	)

Parameters

x	Random variate \( x \)
alpha	Shape \( \alpha > 0 \)
beta	Shape \( \beta > 0 \)

Returns: \( f(x) \) where \( f \) is the probability density function of a beta random variable with shape parameters \( \alpha \) and \( \beta \)

Definition at line 207 of file prob.sql_in.

float8 beta_quantile	(	float8	p,
		float8	alpha,
		float8	beta
	)

Parameters

p	Probability \( p \in [0,1] \)
alpha	Shape \( \alpha > 0 \)
beta	Shape \( \beta > 0 \)

Returns: \( x \) such that \( p = \Pr[X \leq x] \) where \( X \) is beta distribution random variable with shape parameters \( \alpha \) and \( \beta \)

Definition at line 226 of file prob.sql_in.

float8 binomial_cdf	(	float8	x,
		int4	n,
		float8	sp
	)

Parameters

x	Random variate \( x \)
n	The number of trials \( n \in \mathbb N_0 \)
sp	Success probability \( \mathit{sp} \in [0,1] \)

Returns: \( \Pr[X \leq x] \) where \( X \) is a binomially distributed random variable with \( n \) trials and success probability \( \mathit{sp} \)

Definition at line 246 of file prob.sql_in.

float8 binomial_pmf	(	int4	x,
		int4	n,
		float8	sp
	)

Parameters

x	Random variate \( x \)
n	The number of trials \( n \in \mathbb N_0 \)
sp	Success probability \( \mathit{sp} \in [0,1] \)

Returns: \( f(x) \) where \( f \) is the probability mass function of a binomially distributed random variable with \( n \) trials and success probability \( \mathit{sp} \)

Definition at line 265 of file prob.sql_in.

float8 binomial_quantile	(	float8	p,
		int4	n,
		float8	sp
	)

Parameters

p	Probability \( p \in [0,1] \)
n	The number of trials \( n \in \mathbb N_0 \)
sp	Success probability \( \mathit{sp} \in [0,1] \)

Returns: If \( p < 0.5 \) the maximum \( x \) such that \( p \geq \Pr[X \leq x] \). If \( p \geq 0.5 \) the minimum \( x \) such that \( p \leq \Pr[X \leq x] \). Here, \( X \) is a binomially distributed random variable with \( n \) trials and success probability \( \mathit{sp} \).

Definition at line 286 of file prob.sql_in.

float8 cauchy_cdf	(	float8	x,
		float8	location,
		float8	scale
	)

Parameters

x	Random variate \( x \)
location	Location \( x_0 \)
scale	Scale \( \gamma > 0 \)

Returns: \( \Pr[X \leq x] \) where \( X \) is a Cauchy-distributed random variable with location and scale parameters \( x_0 \) and \( \gamma \), respectively

Definition at line 306 of file prob.sql_in.

float8 cauchy_pdf	(	float8	x,
		float8	location,
		float8	scale
	)

Parameters

x	Random variate \( x \)
location	Location \( x_0 \)
scale	Scale \( \gamma > 0 \)

Returns: \( f(x) \) where \( f \) is the probability density function of a Cauchy-distributed random variable with location and scale parameters \( x_0 \) and \( \gamma \), respectively

Definition at line 325 of file prob.sql_in.

float8 cauchy_quantile	(	float8	p,
		float8	location,
		float8	scale
	)

Parameters

p	Probability \( p \in [0,1] \)
location	Location \( x_0 \)
scale	Scale \( \gamma > 0 \)

Returns: \( x \) such that \( p = \Pr[X \leq x] \) where \( X \) is a Cauchy-distributed random variable with location and scale parameters \( x_0 \) and \( \gamma \), respectively

Definition at line 344 of file prob.sql_in.

float8 chi_squared_cdf	(	float8	x,
		float8	df
	)

Parameters

x	Random variate \( x \)
df	Degrees of freedom \( \nu > 0 \)

Returns: \( \Pr[X \leq x] \) where \( X \) is a chi-squared distributed random variable with \( \nu \) degrees of freedom

Definition at line 362 of file prob.sql_in.

float8 chi_squared_pdf	(	float8	x,
		float8	df
	)

Parameters

x	Random variate \( x \)
df	Degrees of freedom \( \nu > 0 \)

Returns: \( f(x) \) where \( f \) is the probability density function of a chi-squared distributed random variable with \( \nu \) degrees of freedom

Definition at line 379 of file prob.sql_in.

float8 chi_squared_quantile	(	float8	p,
		float8	df
	)

Parameters

p	Probability \( p \in [0,1] \)
df	Degrees of freedom \( \mu > 0 \)

Returns: \( x \) such that \( p = \Pr[X \leq x] \) where \( X \) is a chi-squared distributed random variable with \( \nu \) degrees of freedom

Definition at line 396 of file prob.sql_in.

float8 exponential_cdf	(	float8	x,
		float8	lambda
	)

Parameters

x	Random variate \( x \)
lambda	Rate parameter \( \lambda > 0 \)

Returns: \( \Pr[X \leq x] \) where \( X \) is an exponentially distributed random variable with rate parameter \( \lambda \)

Definition at line 413 of file prob.sql_in.

float8 exponential_pdf	(	float8	x,
		float8	lambda
	)

Parameters

x	Random variate \( x \)
lambda	Rate parameter \( \lambda > 0 \)

Returns: \( f(x) \) where \( f \) is the probability density function of exponentially distributed random variable with rate parameter \( \lambda \)

Definition at line 430 of file prob.sql_in.

float8 exponential_quantile	(	float8	p,
		float8	lambda
	)

Parameters

p	Probability \( p \in [0,1] \)
lambda	Rate parameter \( \lambda > 0 \)

Returns: \( x \) such that \( p = \Pr[X \leq x] \) where \( X \) is a exponentially distributed random variable with rate parameter \( \lambda \)

Definition at line 447 of file prob.sql_in.

float8 extreme_value_cdf	(	float8	x,
		float8	location,
		float8	scale
	)

Parameters

x	Random variate \( x \)
location	Location \( \alpha \)
scale	Scale \( \beta > 0 \)

Returns: \( \Pr[X \leq x] \) where \( X \) is an extreme-value distributed random variable with location and scale parameters \( \alpha \) and \( \beta \), respectively

Definition at line 466 of file prob.sql_in.

float8 extreme_value_pdf	(	float8	x,
		float8	location,
		float8	scale
	)

Parameters

x	Random variate \( x \)
location	Location \( \alpha \)
scale	Scale \( \beta > 0 \)

Returns: \( f(x) \) where \( f \) is the probability density function of an extreme-value distributed random variable with location and scale parameters \( \alpha \) and \( \beta \), respectively

Definition at line 485 of file prob.sql_in.

float8 extreme_value_quantile	(	float8	p,
		float8	location,
		float8	scale
	)

Parameters

p	Probability \( p \in [0,1] \)
location	Location \( \alpha \)
scale	Scale \( \beta > 0 \)

Returns: \( x \) such that \( p = \Pr[X \leq x] \) where \( X \) is an extreme-value distributed random variable with location and scale parameters \( \alpha \) and \( \beta \), respectively

Definition at line 504 of file prob.sql_in.

float8 fisher_f_cdf	(	float8	x,
		float8	df1,
		float8	df2
	)

Parameters

x	Random variate \( x \)
df1	Degrees of freedom in numerator \( \nu_1 > 0 \)
df2	Degrees of freedom in denominator \( \nu_1 > 0 \)

Returns: \( \Pr[X \leq x] \) where \( X \) is a Fisher F-distributed random variable with parameters \( \nu_1 \) and \( \nu_2 \)

Definition at line 523 of file prob.sql_in.

float8 fisher_f_pdf	(	float8	x,
		float8	df1,
		float8	df2
	)

Parameters

x	Random variate \( x \)
df1	Degrees of freedom in numerator \( \nu_1 > 0 \)
df2	Degrees of freedom in denominator \( \nu_1 > 0 \)

Returns: \( f(x) \) where \( f \) is the probability density function of a Fisher F-distributed random variable with parameters \( \nu_1 \) and \( \nu_2 \)

Definition at line 542 of file prob.sql_in.

float8 fisher_f_quantile	(	float8	p,
		float8	df1,
		float8	df2
	)

Parameters

p	Probability \( p \in [0,1] \)
df1	Degrees of freedom in numerator \( \nu_1 > 0 \)
df2	Degrees of freedom in denominator \( \nu_1 > 0 \)

Returns: \( x \) such that \( p = \Pr[X \leq x] \) where \( X \) is a Fisher F-distributed random variable with parameters \( \nu_1 \) and \( \nu_2 \)

Definition at line 561 of file prob.sql_in.

float8 gamma_cdf	(	float8	x,
		float8	shape,
		float8	scale
	)

Parameters

x	Random variate \( x \)
shape	Shape \( k > 0 \)
scale	Scale \( \theta > 0 \)

Returns: \( \Pr[X \leq x] \) where \( X \) is a gamma distributed random variable with shape and scale parameters \( k \) and \( \theta \), respectively

Definition at line 581 of file prob.sql_in.

float8 gamma_pdf	(	float8	x,
		float8	shape,
		float8	scale
	)

Parameters

x	Random variate \( x \)
shape	Shape \( k > 0 \)
scale	Scale \( \theta > 0 \)

Returns: \( f(x) \) where \( f \) is the probability density function of a gamma distributed random variable with shape and scale parameters \( k \) and \( \theta \), respectively

Definition at line 600 of file prob.sql_in.

float8 gamma_quantile	(	float8	p,
		float8	shape,
		float8	scale
	)

Parameters

p	Probability \( p \in [0,1] \)
shape	Shape \( k > 0 \)
scale	Scale \( \theta > 0 \)

Returns: \( x \) such that \( p = \Pr[X \leq x] \) where \( X \) is a gamma distributed random variable with shape and scale parameters \( k \) and \( \theta \), respectively

Definition at line 619 of file prob.sql_in.

float8 geometric_cdf	(	float8	x,
		float8	sp
	)

Parameters

x	Random variate \( x \)
sp	Success probability \( \mathit{sp} \in [0,1] \)

Returns: \( \Pr[X \leq x] \) where \( X \) is a geometrically distributed random variable with success probability \( \mathit{sp} \).

Definition at line 637 of file prob.sql_in.

float8 geometric_pmf	(	int4	x,
		float8	sp
	)

Parameters

x	Random variate \( x \)
sp	Success probability \( \mathit{sp} \in [0,1] \)

Returns: \( f(x) \) where \( f \) is the probability mass function of a geometrically distributed random variable with success probability \( \mathit{sp} \)

Definition at line 654 of file prob.sql_in.

float8 geometric_quantile	(	float8	p,
		float8	sp
	)

Parameters

p	Probability \( p \in [0,1] \)
sp	Success probability \( \mathit{sp} \in [0,1] \)

Returns: If \( p < 0.5 \) the maximum \( x \) such that \( p \geq \Pr[X \leq x] \). If \( p \geq 0.5 \) the minimum \( x \) such that \( p \leq \Pr[X \leq x] \). Here, \( X \) is a geometrically distributed random variable with success probability \( \mathit{sp} \).

Definition at line 673 of file prob.sql_in.

float8 hypergeometric_cdf	(	float8	x,
		int4	r,
		int4	n,
		int4	N
	)

Parameters

x	Random variate \( x \)
r	Number \( r \in \{ 0, 1, \dots, N \} \) of items with distinct property (sometimes called the number of success states in population)
n	Number \( n \in \{ 0, 1, \dots, N \} \) of draws (without replacement)
N	Total number \( N \in \mathbb N \) of items

Returns: \( \Pr[X \leq x] \) where \( X \) is a hypergeometrically distributed random variable with parameters \( r, n, N \)

Definition at line 698 of file prob.sql_in.

float8 hypergeometric_pmf	(	int4	x,
		int4	r,
		int4	n,
		int4	N
	)

Parameters

x	Random variate \( x \)
r	Number \( r \in \{ 0, 1, \dots, N \} \) of items with distinct property (sometimes called the number of success states in population)
n	Number \( n \in \{ 0, 1, \dots, N \} \) of draws (without replacement)
N	Total number \( N \in \mathbb N \) of items

Returns: \( f(x) \) where \( f \) is the probability mass function of a hypergeometrically distributed random variable with parameters \( r, n, N \)

Definition at line 722 of file prob.sql_in.

float8 hypergeometric_quantile	(	float8	p,
		int4	r,
		int4	n,
		int4	N
	)

Parameters

p	Probability \( p \in [0,1] \)
r	Number \( r \in \{ 0, 1, \dots, N \} \) of items with distinct property (sometimes called the number of success states in population)
n	Number \( n \in \{ 0, 1, \dots, N \} \) of draws (without replacement)
N	Total number \( N \in \mathbb N \) of items

Returns: \( x \) such that \( p = \Pr[X \leq x] \) where \( X \) is a hypergeometrically distributed random variable with parameters \( r, n, N \)

Definition at line 747 of file prob.sql_in.

float8 inverse_gamma_cdf	(	float8	x,
		float8	shape,
		float8	scale
	)

Parameters

x	Random variate \( x \)
shape	Shape \( \alpha > 0 \)
scale	Scale \( \beta > 0 \)

Returns: \( \Pr[X \leq x] \) where \( X \) is an inverse-gamma distributed random variable with shape and scale parameters \( \alpha \) and \( \beta \), respectively

Definition at line 768 of file prob.sql_in.

float8 inverse_gamma_pdf	(	float8	x,
		float8	shape,
		float8	scale
	)

Parameters

x	Random variate \( x \)
shape	Shape \( \alpha > 0 \)
scale	Scale \( \beta > 0 \)

Returns: \( f(x) \) where \( f \) is the probability density function of an inverse-gamma distributed random variable with shape and scale parameters \( \alpha \) and \( \beta \), respectively

Definition at line 787 of file prob.sql_in.

float8 inverse_gamma_quantile	(	float8	p,
		float8	shape,
		float8	scale
	)

Parameters

p	Probability \( p \in [0,1] \)
shape	Shape \( \alpha > 0 \)
scale	Scale \( \beta > 0 \)

Returns: \( x \) such that \( p = \Pr[X \leq x] \) where \( X \) is an inverse-gamma distributed random variable with shape and scale parameters \( \alpha \) and \( \beta \), respectively

Definition at line 806 of file prob.sql_in.

float8 kolmogorov_cdf ( float8 x)

Parameters

x	Random variate \( x \)

Returns: \( \Pr[X \leq x] \) where \( X \) is a Kolmogorov distributed random variable

See Also: Kolmogorov-Smirnov test: ks_test()

Definition at line 825 of file prob.sql_in.

float8 laplace_cdf	(	float8	x,
		float8	mean,
		float8	scale
	)

Parameters

x	Random variate \( x \)
mean	Mean \( \mu \)
scale	Scale \( b > 0 \)

Returns: \( \Pr[X \leq x] \) where \( X \) is a Laplace-distributed random variable with mean \( \mu \) and variance \( 2 b^2 \)

Definition at line 842 of file prob.sql_in.

float8 laplace_pdf	(	float8	x,
		float8	mean,
		float8	scale
	)

Parameters

x	Random variate \( x \)
mean	Mean \( \mu \)
scale	Scale \( b > 0 \)

Returns: \( f(x) \) where \( f \) is the probability density function of a Laplace-distributed random variable with mean \( \mu \) and variance \( 2 b^2 \)

Definition at line 861 of file prob.sql_in.

float8 laplace_quantile	(	float8	p,
		float8	mean,
		float8	scale
	)

Parameters

p	Probability \( p \in [0,1] \)
mean	Mean \( \mu \)
scale	Scale \( b > 0 \)

Returns: \( x \) such that \( p = \Pr[X \leq x] \) where \( X \) is a Laplace-distributed random variable with mean \( \mu \) and variance \( 2 b^2 \)

Definition at line 880 of file prob.sql_in.

float8 logistic_cdf	(	float8	x,
		float8	mean,
		float8	scale
	)

Parameters

x	Random variate \( x \)
mean	Mean \( \mu \)
scale	Scale \( s > 0 \)

Returns: \( \Pr[X \leq x] \) where \( X \) is a logistically distributed random variable with mean \( \mu \) and scale parameter \( s \)

Definition at line 899 of file prob.sql_in.

float8 logistic_pdf	(	float8	x,
		float8	mean,
		float8	scale
	)

Parameters

x	Random variate \( x \)
mean	Mean \( \mu \)
scale	Scale \( s > 0 \)

Returns: \( f(x) \) where \( f \) is the probability density function of a logistically distributed random variable with mean \( \mu \) and scale parameter \( s \)

Definition at line 918 of file prob.sql_in.

float8 logistic_quantile	(	float8	p,
		float8	mean,
		float8	scale
	)

Parameters

p	Probability \( p \in [0,1] \)
mean	Mean \( \mu \)
scale	Scale \( s > 0 \)

Returns: \( x \) such that \( p = \Pr[X \leq x] \) where \( X \) is a logistically distributed random variable with mean \( \mu \) and scale parameter \( s \)

Definition at line 937 of file prob.sql_in.

float8 lognormal_cdf	(	float8	x,
		float8	location,
		float8	scale
	)

Parameters

x	Random variate \( x \)
location	Location \( m \)
scale	Scale \( s > 0 \)

Returns: \( \Pr[X \leq x] \) where \( X \) is a lognormally distributed random variable with location and scale parameters \( m \) and \( s \), respectively

Definition at line 957 of file prob.sql_in.

float8 lognormal_pdf	(	float8	x,
		float8	location,
		float8	scale
	)

Parameters

x	Random variate \( x \)
location	Location \( m \)
scale	Scale \( s > 0 \)

Returns: \( f(x) \) where \( f \) is the probability density function of a lognormally distributed random variable with location and scale parameters \( m \) and \( s \), respectively

Definition at line 976 of file prob.sql_in.

float8 lognormal_quantile	(	float8	p,
		float8	location,
		float8	scale
	)

Parameters

p	Probability \( p \in [0,1] \)
location	Location \( m \)
scale	Scale \( s > 0 \)

Returns: \( x \) such that \( p = \Pr[X \leq x] \) where \( X \) is a lognormally distributed random variable with location and scale parameters \( m \) and \( s \), respectively

Definition at line 995 of file prob.sql_in.

float8 negative_binomial_cdf	(	float8	x,
		float8	r,
		float8	sp
	)

Parameters

x	Random variate \( x \)
r	Total number \( r > 0 \) of successes in \( x + r \) trials (assuming success in the last trial)
sp	Success probability \( \mathit{sp} \in (0,1] \) in each trial

Returns: \( \Pr[X \leq x] \) where \( X \) is a negative-binomially distributed random variable with parameters \( r, \mathit{sp} \)

Definition at line 1015 of file prob.sql_in.

float8 negative_binomial_pmf	(	int4	x,
		float8	r,
		float8	sp
	)

Parameters

x	Random variate \( x \)
r	Total number \( r > 0 \) of successes in \( x + r \) trials (assuming success in the last trial)
sp	Success probability \( \mathit{sp} \in (0,1] \) in each trial

Returns: \( f(x) \) where \( f \) is the probability mass function of a negative-binomially distributed random variable with parameters \( r, \mathit{sp} \)

Definition at line 1035 of file prob.sql_in.

float8 negative_binomial_quantile	(	float8	p,
		float8	r,
		float8	sp
	)

Parameters

p	Probability \( p \in [0,1] \)
r	Total number \( r > 0 \) of successes in \( x + r \) trials (assuming success in the last trial)
sp	Success probability \( \mathit{sp} \in (0,1] \) in each trial

Returns: If \( p < 0.5 \) the maximum \( x \) such that \( p \geq \Pr[X \leq x] \). If \( p \geq 0.5 \) the minimum \( x \) such that \( p \leq \Pr[X \leq x] \). Here, \( X \) is a negative-binomially distributed random variable with parameters \( r, \mathit{sp} \)

Definition at line 1057 of file prob.sql_in.

float8 non_central_beta_cdf	(	float8	x,
		float8	alpha,
		float8	beta,
		float8	ncp
	)

Parameters

x	Random variate \( x \)
alpha	Shape \( \alpha > 0 \)
beta	Shape \( \beta > 0 \)
ncp	Noncentrality parameter \( \delta \geq 0 \)

Returns: \( \Pr[X \leq x] \) where \( X \) is a noncentral-beta distributed random variable with shape parameters \( shape_1 \) and \( shape_2 \) and noncentrality parameter \( \delta \)

Definition at line 1078 of file prob.sql_in.

float8 non_central_beta_pdf	(	float8	x,
		float8	alpha,
		float8	beta,
		float8	ncp
	)

Parameters

x	Random variate \( x \)
alpha	Shape \( \alpha > 0 \)
beta	Shape \( \beta > 0 \)
ncp	Noncentrality parameter \( \delta \geq 0 \)

Returns: \( f(x) \) where \( f \) is the probability density function of a noncentral-beta distributed random variable with shape parameters \( shape_1 \) and \( shape_2 \) and noncentrality parameter \( \delta \)

Definition at line 1100 of file prob.sql_in.

float8 non_central_beta_quantile	(	float8	p,
		float8	alpha,
		float8	beta,
		float8	ncp
	)

Parameters

p	Probability \( p \in [0,1] \)
alpha	Shape \( \alpha > 0 \)
beta	Shape \( \beta > 0 \)
ncp	Noncentrality parameter \( \delta \geq 0 \)

Returns: \( x \) such that \( p = \Pr[X \leq x] \) where \( X \) is a noncentral-beta distributed random variable with shape parameters \( shape_1 \) and \( shape_2 \) and noncentrality parameter \( \delta \)

Definition at line 1122 of file prob.sql_in.

float8 non_central_chi_squared_cdf	(	float8	x,
		float8	df,
		float8	ncp
	)

Parameters

x	Random variate \( x \)
df	Degrees of freedom \( \nu > 0 \)
ncp	The noncentrality parameter \( \lambda \geq 0 \)

Returns: \( \Pr[X \leq x] \) where \( X \) is a noncentral-chi-squared distributed random variable with \( \nu \) degrees of freedom and noncentrality parameter \( \lambda \)

Definition at line 1143 of file prob.sql_in.

float8 non_central_chi_squared_pdf	(	float8	x,
		float8	df,
		float8	ncp
	)

Parameters

x	Random variate \( x \)
df	Degrees of freedom \( \nu > 0 \)
ncp	The noncentrality parameter \( \lambda \geq 0 \)

Returns: \( f(x) \) where \( f \) is the probability density function of a noncentral-chi-squared distributed random variable with \( \nu \) degrees of freedom and noncentrality parameter \( \lambda \)

Definition at line 1162 of file prob.sql_in.

float8 non_central_chi_squared_quantile	(	float8	p,
		float8	df,
		float8	ncp
	)

Parameters

p	Probability \( p \in [0,1] \)
df	Degrees of freedom \( \nu > 0 \)
ncp	The noncentrality parameter \( \lambda \geq 0 \)

Returns: \( x \) such that \( p = \Pr[X \leq x] \) where \( X \) is a noncentral-chi-squared distributed random variable with \( \nu \) degrees of freedom and noncentrality parameter \( \lambda \)

Definition at line 1181 of file prob.sql_in.

float8 non_central_f_cdf	(	float8	x,
		float8	df1,
		float8	df2,
		float8	ncp
	)

Parameters

x	Random variate \( x \)
df1	Degrees of freedom in numerator \( \nu_1 > 0 \)
df2	Degrees of freedom in denominator \( \nu_1 > 0 \)
ncp	The noncentrality parameter \( \lambda \geq 0 \)

Returns: \( \Pr[X \leq x] \) where \( X \) is a noncentral Fisher F-distributed random variable with parameters \( \nu_1, \nu_2, \lambda \)

Definition at line 1202 of file prob.sql_in.

float8 non_central_f_pdf	(	float8	x,
		float8	df1,
		float8	df2,
		float8	ncp
	)

Parameters

x	Random variate \( x \)
df1	Degrees of freedom in numerator \( \nu_1 > 0 \)
df2	Degrees of freedom in denominator \( \nu_1 > 0 \)
ncp	The noncentrality parameter \( \lambda \geq 0 \)

Returns: \( f(x) \) where \( f \) is the probability density function of a noncentral Fisher F-distributed random variable with parameters \( \nu_1, \nu_2, \lambda \)

Definition at line 1223 of file prob.sql_in.

float8 non_central_f_quantile	(	float8	p,
		float8	df1,
		float8	df2,
		float8	ncp
	)

Parameters

p	Probability \( p \in [0,1] \)
df1	Degrees of freedom in numerator \( \nu_1 > 0 \)
df2	Degrees of freedom in denominator \( \nu_1 > 0 \)
ncp	The noncentrality parameter \( \lambda \geq 0 \)

Returns: \( x \) such that \( p = \Pr[X \leq x] \) where \( X \) is a noncentral Fisher F-distributed random variable with parameters \( \nu_1, \nu_2, \lambda \)

Definition at line 1244 of file prob.sql_in.

float8 non_central_t_cdf	(	float8	x,
		float8	df,
		float8	ncp
	)

Parameters

x	Random variate \( x \)
df	Degrees of freedom \( \nu > 0 \)
ncp	Noncentrality parameter \( \delta \)

Returns: \( \Pr[X \leq x] \) where \( X \) is a noncentral Student's t-distributed random variable with \( \nu \) degrees of freedom and noncentrality parameter \( \delta \)

Definition at line 1265 of file prob.sql_in.

float8 non_central_t_pdf	(	float8	x,
		float8	df,
		float8	ncp
	)

Parameters

x	Random variate \( x \)
df	Degrees of freedom \( \nu > 0 \)
ncp	Noncentrality parameter \( \delta \)

Returns: \( x \) such that \( p = \Pr[X \leq x] \) where \( X \) is a noncentral Student's t-distributed random variable with \( \nu \) degrees of freedom and noncentrality parameter \( \delta \)

Definition at line 1284 of file prob.sql_in.

float8 non_central_t_quantile	(	float8	p,
		float8	df,
		float8	ncp
	)

Parameters

p	Probability \( p \in [0,1] \)
df	Degrees of freedom \( \nu > 0 \)
ncp	Noncentrality parameter \( \delta \)

Returns: \( x \) such that \( p = \Pr[X \leq x] \) where \( X \) is a noncentral Student's t-distributed random variable with \( \nu \) degrees of freedom and noncentrality parameter \( \delta \)

Definition at line 1303 of file prob.sql_in.

float8 normal_cdf	(	float8	x,
		float8	mean = `0`,
		float8	sd = `1`
	)

Parameters

x	Random variate \( x \)
mean	Mean \( \mu \)
sd	Standard deviation \( \sigma > 0 \)

Returns: \( \Pr[X \leq x] \) where \( T \) is a normally distributed random variable with mean \( \mu \) and variance \( \sigma^2 \)

Definition at line 1322 of file prob.sql_in.

float8 normal_pdf	(	float8	x,
		float8	mean = `0`,
		float8	sd = `1`
	)

Parameters

x	Random variate \( x \)
mean	Mean \( \mu \)
sd	Standard deviation \( \sigma > 0 \)

Returns: \( f(x) \) where \( f \) is the probability density function of a normally distributed random variable with mean \( \mu \) and variance \( \sigma^2 \)

Definition at line 1360 of file prob.sql_in.

float8 normal_quantile	(	float8	p,
		float8	mean = `0`,
		float8	sd = `1`
	)

Parameters

p	Probability \( p \in [0,1] \)
mean	Mean \( \mu \)
sd	Standard deviation \( \sigma > 0 \)

Returns: \( x \) such that \( p = \Pr[X \leq x] \) where \( X \) is a normally distributed random variable with mean \( \mu \) and variance \( \sigma^2 \)

Definition at line 1398 of file prob.sql_in.

float8 pareto_cdf	(	float8	x,
		float8	scale,
		float8	shape
	)

Parameters

x	Random variate \( x \)
scale	Scale \( \beta > 0 \)
shape	Shape \( \alpha > 0 \)

Returns: \( \Pr[X \leq x] \) where \( X \) is a Pareto-distributed random variable with shape and scale parameters \( \alpha \) and \( \beta \), respectively

Definition at line 1437 of file prob.sql_in.

float8 pareto_pdf	(	float8	x,
		float8	scale,
		float8	shape
	)

Parameters

x	Random variate \( x \)
scale	Scale \( \beta > 0 \)
shape	Shape \( \alpha > 0 \)

Returns: \( f(x) \) where \( f \) is the probability density function of a Pareto-distributed random variable with shape and scale parameters \( \alpha \) and \( \beta \), respectively

Definition at line 1456 of file prob.sql_in.

float8 pareto_quantile	(	float8	p,
		float8	scale,
		float8	shape
	)

Parameters

p	Probability \( p \in [0,1] \)
scale	Scale \( \beta > 0 \)
shape	Shape \( \alpha > 0 \)

Returns: \( x \) such that \( p = \Pr[X \leq x] \) where \( X \) is a Pareto-distributed random variable with shape and scale parameters \( \alpha \) and \( \beta \), respectively

Definition at line 1475 of file prob.sql_in.

float8 poisson_cdf	(	float8	x,
		float8	mean
	)

Parameters

x	Random variate \( x \)
mean	Average occurrence rate \( \lambda > 0 \)

Returns: \( \Pr[X \leq x] \) where \( X \) is a Poisson distributed random variable with mean \( \lambda \)

Definition at line 1493 of file prob.sql_in.

float8 poisson_pmf	(	int4	x,
		float8	mean
	)

Parameters

x	Random variate \( x \)
mean	Average occurrence rate \( \lambda > 0 \)

Returns: \( f(x) \) where \( f \) is the probability mass function of a Poisson distributed random variable with mean \( \lambda \)

Definition at line 1509 of file prob.sql_in.

float8 poisson_quantile	(	float8	p,
		float8	mean
	)

Parameters

p	Probability \( p \in [0,1] \)
mean	Average occurrence rate \( \lambda > 0 \)

Returns: If \( p < 0.5 \) the maximum \( x \) such that \( p \geq \Pr[X \leq x] \). If \( p \geq 0.5 \) the minimum \( x \) such that \( p \leq \Pr[X \leq x] \). Here, \( X \) is a Poisson distributed random variable with mean \( \lambda \)

Definition at line 1527 of file prob.sql_in.

float8 rayleigh_cdf	(	float8	x,
		float8	scale
	)

Parameters

x	Random variate \( x \)
scale	Scale \( \sigma > 0 \)

Returns: \( \Pr[X \leq x] \) where \( X \) is a Rayleigh-distributed random variable with parameter \( \sigma \)

Definition at line 1544 of file prob.sql_in.

float8 rayleigh_pdf	(	float8	x,
		float8	scale
	)

Parameters

x	Random variate \( x \)
scale	Scale \( \sigma > 0 \)

Returns: \( f(x) \) where \( f \) is the probability density function of a Rayleigh-distributed random variable with parameter \( \sigma \)

Definition at line 1560 of file prob.sql_in.

float8 rayleigh_quantile	(	float8	p,
		float8	scale
	)

Parameters

p	Probability \( p \in [0,1] \)
scale	Scale \( \sigma > 0 \)

Returns: \( x \) such that \( p = \Pr[X \leq x] \) where \( X \) is a Rayleigh-distributed random variable with parameter \( \sigma \)

Definition at line 1576 of file prob.sql_in.

float8 students_t_cdf	(	float8	x,
		float8	df
	)

Parameters

x	Random variate \( x \)
df	Degrees of freedom \( \nu > 0 \)

Returns: \( \Pr[X \leq x] \) where \( X \) is a Student's t-distributed random variable with \( \nu \) degrees of freedom

Definition at line 1593 of file prob.sql_in.

float8 students_t_pdf	(	float8	x,
		float8	df
	)

Parameters

x	Random variate \( x \)
df	Degrees of freedom \( \nu > 0 \)

Returns: \( f(x) \) where \( f \) is the probability density function of a Stundent's t-distributed random variable with \( \nu \) degrees of freedom

Definition at line 1610 of file prob.sql_in.

float8 students_t_quantile	(	float8	p,
		float8	df
	)

Parameters

p	Probability \( p \in [0,1] \)
df	Degrees of freedom \( \nu > 0 \)

Returns: \( x \) such that \( p = \Pr[X \leq x] \) where \( X \) is a Student's t-distributed random variable with \( \nu \) degrees of freedom

Definition at line 1627 of file prob.sql_in.

float8 triangular_cdf	(	float8	x,
		float8	lower,
		float8	mode,
		float8	upper
	)

Parameters

x	Random variate \( x \)
lower	Lower bound \( a \)
mode	Mode \( c \geq a \)
upper	Upper bound \( b \geq c \), where \( b > a \)

Returns: \( \Pr[X \leq x] \) where \( X \) is a triangular distributed random variable with parameters \( a, b, c \)

Definition at line 1646 of file prob.sql_in.

float8 triangular_pdf	(	float8	x,
		float8	lower,
		float8	mode,
		float8	upper
	)

Parameters

x	Random variate \( x \)
lower	Lower bound \( a \)
mode	Mode \( c \geq a \)
upper	Upper bound \( b \geq c \), where \( b > a \)

Returns: \( f(x) \) where \( f \) is the probability density function of a triangular distributed random variable with parameters \( a, b, c \)

Definition at line 1666 of file prob.sql_in.

float8 triangular_quantile	(	float8	p,
		float8	lower,
		float8	mode,
		float8	upper
	)

Parameters

p	Probability \( p \in [0,1] \)
lower	Lower bound \( a \)
mode	Mode \( c \geq a \)
upper	Upper bound \( b \geq c \), where \( b > a \)

Returns: \( x \) such that \( p = \Pr[X \leq x] \) where \( X \) is a trianbular distributed random variable with parameters \( a, b, c \)

Definition at line 1686 of file prob.sql_in.

float8 uniform_cdf	(	float8	x,
		float8	lower,
		float8	upper
	)

Parameters

x	Random variate \( x \)
lower	Lower bound \( a \)
upper	Upper bound \( b \)

Returns: \( \Pr[X \leq x] \) where \( X \) is a uniform distributed random variable with support \( [a, b] \)

Definition at line 1706 of file prob.sql_in.

float8 uniform_pdf	(	float8	x,
		float8	lower,
		float8	upper
	)

Parameters

x	Random variate \( x \)
lower	Lower bound \( a \)
upper	Upper bound \( b \)

Returns: \( f(x) \) where \( f \) is the probability density function of a uniform distributed random variable with support \( [a, b] \)

Definition at line 1724 of file prob.sql_in.

float8 uniform_quantile	(	float8	p,
		float8	lower,
		float8	upper
	)

Parameters

p	Probability \( p \in [0,1] \)
lower	Lower bound \( a \)
upper	Upper bound \( b \)

Returns: \( x \) such that \( p = \Pr[X \leq x] \) where \( X \) is a uniform distributed random variable with support \( [a, b] \)

Definition at line 1742 of file prob.sql_in.

float8 weibull_cdf	(	float8	x,
		float8	shape,
		float8	scale
	)

Parameters

x	Random variate \( x \)
shape	Shape \( \alpha > 0 \)
scale	Scale \( \beta > 0 \)

Returns: \( \Pr[X \leq x] \) where \( X \) is a weibull distributed random variable with shape and scale parameters \( \alpha \) and \( \beta \), respectively

Definition at line 1762 of file prob.sql_in.

float8 weibull_pdf	(	float8	x,
		float8	shape,
		float8	scale
	)

Parameters

x	Random variate \( x \)
shape	Shape \( \alpha > 0 \)
scale	Scale \( \beta > 0 \)

Returns: \( f(x) \) where \( f \) is the probability density function of a weibull distributed random variable with shape and scale parameters \( \alpha \) and \( \beta \), respectively

Definition at line 1781 of file prob.sql_in.

float8 weibull_quantile	(	float8	p,
		float8	shape,
		float8	scale
	)

Parameters

p	Probability \( p \in [0,1] \)
shape	Shape \( \alpha > 0 \)
scale	Scale \( \beta > 0 \)

Returns: \( x \) such that \( p = \Pr[X \leq x] \) where \( X \) is a weibull distributed random variable with shape and scale parameters \( \alpha \) and \( \beta \), respectively

Definition at line 1800 of file prob.sql_in.

Functions

Detailed Description

Function Documentation