Distribution functions · phit

Densities, masses, tails, and quantiles of 12 families in 13 parameterizations, each in a scalar form and an _array batch form, with a seeded sampler. Inputs, results, and error codes follow the shared rules; families and domains are in Probability distributions.

Normal

Parameter	Type	Default	Description
`$mean`	`float`	`0.0`	Mean
`$sd`	`float`	`1.0`	Standard deviation, > 0

Function	Operation	Arguments	Returns
`dist_normal_pdf()`	Density	`float $x`, parameters	`float`
`dist_normal_pdf_array()`	Density over a list	`array $values`, parameters	`array`
`dist_normal_logpdf()`	Log density	`float $x`, parameters	`float`
`dist_normal_logpdf_array()`	Log density over a list	`array $values`, parameters	`array`
`dist_normal_cdf()`	Lower tail P(X ≤ x)	`float $x`, parameters	`float`
`dist_normal_cdf_array()`	Lower tail P(X ≤ x) over a list	`array $values`, parameters	`array`
`dist_normal_logcdf()`	Log lower tail	`float $x`, parameters	`float`
`dist_normal_logcdf_array()`	Log lower tail over a list	`array $values`, parameters	`array`
`dist_normal_sf()`	Upper tail P(X > x)	`float $x`, parameters	`float`
`dist_normal_sf_array()`	Upper tail P(X > x) over a list	`array $values`, parameters	`array`
`dist_normal_logsf()`	Log upper tail	`float $x`, parameters	`float`
`dist_normal_logsf_array()`	Log upper tail over a list	`array $values`, parameters	`array`
`dist_normal_quantile()`	Quantile	`float $p`, parameters	`float`
`dist_normal_quantile_array()`	Quantile over a list	`array $probabilities`, parameters	`array`
`dist_normal_rng()`	Seeded samples	`int $n`, `int $seed`, parameters	`array`

Phit\dist_normal_cdf(    float $x,    float $mean = 0.0,    float $sd = 1.0,): floatPhit\dist_normal_cdf_array(    array $values,    float $mean = 0.0,    float $sd = 1.0,): array

Errors Exception: code 1001 for an invalid parameter, 1003 for a NAN input, or 1006 for a numerical failure.

Sampler errors Exception: code 1001 for invalid input, 1003 for a nonfinite parameter or sample, or 1007 for more than 1,000,000 draws or an unsupported size.

Student’s t

Parameter	Type	Default	Description
`$df`	`float`	required	Degrees of freedom, > 0

Function	Operation	Arguments	Returns
`dist_t_pdf()`	Density	`float $x`, parameters	`float`
`dist_t_pdf_array()`	Density over a list	`array $values`, parameters	`array`
`dist_t_logpdf()`	Log density	`float $x`, parameters	`float`
`dist_t_logpdf_array()`	Log density over a list	`array $values`, parameters	`array`
`dist_t_cdf()`	Lower tail P(X ≤ x)	`float $x`, parameters	`float`
`dist_t_cdf_array()`	Lower tail P(X ≤ x) over a list	`array $values`, parameters	`array`
`dist_t_logcdf()`	Log lower tail	`float $x`, parameters	`float`
`dist_t_logcdf_array()`	Log lower tail over a list	`array $values`, parameters	`array`
`dist_t_sf()`	Upper tail P(X > x)	`float $x`, parameters	`float`
`dist_t_sf_array()`	Upper tail P(X > x) over a list	`array $values`, parameters	`array`
`dist_t_logsf()`	Log upper tail	`float $x`, parameters	`float`
`dist_t_logsf_array()`	Log upper tail over a list	`array $values`, parameters	`array`
`dist_t_quantile()`	Quantile	`float $p`, parameters	`float`
`dist_t_quantile_array()`	Quantile over a list	`array $probabilities`, parameters	`array`
`dist_t_rng()`	Seeded samples	`int $n`, `int $seed`, parameters	`array`

Phit\dist_t_cdf(    float $x,    float $df,): floatPhit\dist_t_cdf_array(    array $values,    float $df,): array

Errors Exception: code 1001 for an invalid parameter, 1003 for a NAN input, or 1006 for a numerical failure.

Sampler errors Exception: code 1001 for invalid input, 1003 for a nonfinite parameter or sample, or 1007 for more than 1,000,000 draws or an unsupported size.

Chi-square

Parameter	Type	Default	Description
`$df`	`float`	required	Degrees of freedom, > 0

Function	Operation	Arguments	Returns
`dist_chisquare_pdf()`	Density	`float $x`, parameters	`float`
`dist_chisquare_pdf_array()`	Density over a list	`array $values`, parameters	`array`
`dist_chisquare_logpdf()`	Log density	`float $x`, parameters	`float`
`dist_chisquare_logpdf_array()`	Log density over a list	`array $values`, parameters	`array`
`dist_chisquare_cdf()`	Lower tail P(X ≤ x)	`float $x`, parameters	`float`
`dist_chisquare_cdf_array()`	Lower tail P(X ≤ x) over a list	`array $values`, parameters	`array`
`dist_chisquare_logcdf()`	Log lower tail	`float $x`, parameters	`float`
`dist_chisquare_logcdf_array()`	Log lower tail over a list	`array $values`, parameters	`array`
`dist_chisquare_sf()`	Upper tail P(X > x)	`float $x`, parameters	`float`
`dist_chisquare_sf_array()`	Upper tail P(X > x) over a list	`array $values`, parameters	`array`
`dist_chisquare_logsf()`	Log upper tail	`float $x`, parameters	`float`
`dist_chisquare_logsf_array()`	Log upper tail over a list	`array $values`, parameters	`array`
`dist_chisquare_quantile()`	Quantile	`float $p`, parameters	`float`
`dist_chisquare_quantile_array()`	Quantile over a list	`array $probabilities`, parameters	`array`
`dist_chisquare_rng()`	Seeded samples	`int $n`, `int $seed`, parameters	`array`

Phit\dist_chisquare_cdf(    float $x,    float $df,): floatPhit\dist_chisquare_cdf_array(    array $values,    float $df,): array

Errors Exception: code 1001 for an invalid parameter, 1003 for a NAN input, or 1006 for a numerical failure.

Sampler errors Exception: code 1001 for invalid input, 1003 for a nonfinite parameter or sample, or 1007 for more than 1,000,000 draws or an unsupported size.

F

Parameter	Type	Default	Description
`$df1`	`float`	required	Numerator degrees of freedom, > 0
`$df2`	`float`	required	Denominator degrees of freedom, > 0

Function	Operation	Arguments	Returns
`dist_f_pdf()`	Density	`float $x`, parameters	`float`
`dist_f_pdf_array()`	Density over a list	`array $values`, parameters	`array`
`dist_f_logpdf()`	Log density	`float $x`, parameters	`float`
`dist_f_logpdf_array()`	Log density over a list	`array $values`, parameters	`array`
`dist_f_cdf()`	Lower tail P(X ≤ x)	`float $x`, parameters	`float`
`dist_f_cdf_array()`	Lower tail P(X ≤ x) over a list	`array $values`, parameters	`array`
`dist_f_logcdf()`	Log lower tail	`float $x`, parameters	`float`
`dist_f_logcdf_array()`	Log lower tail over a list	`array $values`, parameters	`array`
`dist_f_sf()`	Upper tail P(X > x)	`float $x`, parameters	`float`
`dist_f_sf_array()`	Upper tail P(X > x) over a list	`array $values`, parameters	`array`
`dist_f_logsf()`	Log upper tail	`float $x`, parameters	`float`
`dist_f_logsf_array()`	Log upper tail over a list	`array $values`, parameters	`array`
`dist_f_quantile()`	Quantile	`float $p`, parameters	`float`
`dist_f_quantile_array()`	Quantile over a list	`array $probabilities`, parameters	`array`
`dist_f_rng()`	Seeded samples	`int $n`, `int $seed`, parameters	`array`

Phit\dist_f_cdf(    float $x,    float $df1,    float $df2,): floatPhit\dist_f_cdf_array(    array $values,    float $df1,    float $df2,): array

Errors Exception: code 1001 for an invalid parameter, 1003 for a NAN input, or 1006 for a numerical failure.

Sampler errors Exception: code 1001 for invalid input, 1003 for a nonfinite parameter or sample, or 1007 for more than 1,000,000 draws or an unsupported size.

Beta

Parameter	Type	Default	Description
`$alpha`	`float`	required	First shape parameter, > 0
`$beta`	`float`	required	Second shape parameter, > 0

Function	Operation	Arguments	Returns
`dist_beta_pdf()`	Density	`float $x`, parameters	`float`
`dist_beta_pdf_array()`	Density over a list	`array $values`, parameters	`array`
`dist_beta_logpdf()`	Log density	`float $x`, parameters	`float`
`dist_beta_logpdf_array()`	Log density over a list	`array $values`, parameters	`array`
`dist_beta_cdf()`	Lower tail P(X ≤ x)	`float $x`, parameters	`float`
`dist_beta_cdf_array()`	Lower tail P(X ≤ x) over a list	`array $values`, parameters	`array`
`dist_beta_logcdf()`	Log lower tail	`float $x`, parameters	`float`
`dist_beta_logcdf_array()`	Log lower tail over a list	`array $values`, parameters	`array`
`dist_beta_sf()`	Upper tail P(X > x)	`float $x`, parameters	`float`
`dist_beta_sf_array()`	Upper tail P(X > x) over a list	`array $values`, parameters	`array`
`dist_beta_logsf()`	Log upper tail	`float $x`, parameters	`float`
`dist_beta_logsf_array()`	Log upper tail over a list	`array $values`, parameters	`array`
`dist_beta_quantile()`	Quantile	`float $p`, parameters	`float`
`dist_beta_quantile_array()`	Quantile over a list	`array $probabilities`, parameters	`array`
`dist_beta_rng()`	Seeded samples	`int $n`, `int $seed`, parameters	`array`

Phit\dist_beta_cdf(    float $x,    float $alpha,    float $beta,): floatPhit\dist_beta_cdf_array(    array $values,    float $alpha,    float $beta,): array

Errors Exception: code 1001 for an invalid parameter, 1003 for a NAN input, or 1006 for a numerical failure.

Sampler errors Exception: code 1001 for invalid input, 1003 for a nonfinite parameter or sample, or 1007 for more than 1,000,000 draws or an unsupported size.

Gamma

Parameter	Type	Default	Description
`$shape`	`float`	required	Shape, > 0
`$scale`	`float`	`1.0`	Scale, > 0; the mean is shape × scale

Function	Operation	Arguments	Returns
`dist_gamma_pdf()`	Density	`float $x`, parameters	`float`
`dist_gamma_pdf_array()`	Density over a list	`array $values`, parameters	`array`
`dist_gamma_logpdf()`	Log density	`float $x`, parameters	`float`
`dist_gamma_logpdf_array()`	Log density over a list	`array $values`, parameters	`array`
`dist_gamma_cdf()`	Lower tail P(X ≤ x)	`float $x`, parameters	`float`
`dist_gamma_cdf_array()`	Lower tail P(X ≤ x) over a list	`array $values`, parameters	`array`
`dist_gamma_logcdf()`	Log lower tail	`float $x`, parameters	`float`
`dist_gamma_logcdf_array()`	Log lower tail over a list	`array $values`, parameters	`array`
`dist_gamma_sf()`	Upper tail P(X > x)	`float $x`, parameters	`float`
`dist_gamma_sf_array()`	Upper tail P(X > x) over a list	`array $values`, parameters	`array`
`dist_gamma_logsf()`	Log upper tail	`float $x`, parameters	`float`
`dist_gamma_logsf_array()`	Log upper tail over a list	`array $values`, parameters	`array`
`dist_gamma_quantile()`	Quantile	`float $p`, parameters	`float`
`dist_gamma_quantile_array()`	Quantile over a list	`array $probabilities`, parameters	`array`
`dist_gamma_rng()`	Seeded samples	`int $n`, `int $seed`, parameters	`array`

Phit\dist_gamma_cdf(    float $x,    float $shape,    float $scale = 1.0,): floatPhit\dist_gamma_cdf_array(    array $values,    float $shape,    float $scale = 1.0,): array

Errors Exception: code 1001 for an invalid parameter, 1003 for a NAN input, or 1006 for a numerical failure.

Sampler errors Exception: code 1001 for invalid input, 1003 for a nonfinite parameter or sample, or 1007 for more than 1,000,000 draws or an unsupported size.

Exponential

Parameter	Type	Default	Description
`$rate`	`float`	`1.0`	Rate, > 0; the mean is 1 / rate

Function	Operation	Arguments	Returns
`dist_exponential_pdf()`	Density	`float $x`, parameters	`float`
`dist_exponential_pdf_array()`	Density over a list	`array $values`, parameters	`array`
`dist_exponential_logpdf()`	Log density	`float $x`, parameters	`float`
`dist_exponential_logpdf_array()`	Log density over a list	`array $values`, parameters	`array`
`dist_exponential_cdf()`	Lower tail P(X ≤ x)	`float $x`, parameters	`float`
`dist_exponential_cdf_array()`	Lower tail P(X ≤ x) over a list	`array $values`, parameters	`array`
`dist_exponential_logcdf()`	Log lower tail	`float $x`, parameters	`float`
`dist_exponential_logcdf_array()`	Log lower tail over a list	`array $values`, parameters	`array`
`dist_exponential_sf()`	Upper tail P(X > x)	`float $x`, parameters	`float`
`dist_exponential_sf_array()`	Upper tail P(X > x) over a list	`array $values`, parameters	`array`
`dist_exponential_logsf()`	Log upper tail	`float $x`, parameters	`float`
`dist_exponential_logsf_array()`	Log upper tail over a list	`array $values`, parameters	`array`
`dist_exponential_quantile()`	Quantile	`float $p`, parameters	`float`
`dist_exponential_quantile_array()`	Quantile over a list	`array $probabilities`, parameters	`array`
`dist_exponential_rng()`	Seeded samples	`int $n`, `int $seed`, parameters	`array`

Phit\dist_exponential_cdf(    float $x,    float $rate = 1.0,): floatPhit\dist_exponential_cdf_array(    array $values,    float $rate = 1.0,): array

Errors Exception: code 1001 for an invalid parameter, 1003 for a NAN input, or 1006 for a numerical failure.

Sampler errors Exception: code 1001 for invalid input, 1003 for a nonfinite parameter or sample, or 1007 for more than 1,000,000 draws or an unsupported size.

Log-normal

Parameter	Type	Default	Description
`$meanlog`	`float`	`0.0`	Mean of log X
`$sdlog`	`float`	`1.0`	Standard deviation of log X, > 0

Function	Operation	Arguments	Returns
`dist_lognormal_pdf()`	Density	`float $x`, parameters	`float`
`dist_lognormal_pdf_array()`	Density over a list	`array $values`, parameters	`array`
`dist_lognormal_logpdf()`	Log density	`float $x`, parameters	`float`
`dist_lognormal_logpdf_array()`	Log density over a list	`array $values`, parameters	`array`
`dist_lognormal_cdf()`	Lower tail P(X ≤ x)	`float $x`, parameters	`float`
`dist_lognormal_cdf_array()`	Lower tail P(X ≤ x) over a list	`array $values`, parameters	`array`
`dist_lognormal_logcdf()`	Log lower tail	`float $x`, parameters	`float`
`dist_lognormal_logcdf_array()`	Log lower tail over a list	`array $values`, parameters	`array`
`dist_lognormal_sf()`	Upper tail P(X > x)	`float $x`, parameters	`float`
`dist_lognormal_sf_array()`	Upper tail P(X > x) over a list	`array $values`, parameters	`array`
`dist_lognormal_logsf()`	Log upper tail	`float $x`, parameters	`float`
`dist_lognormal_logsf_array()`	Log upper tail over a list	`array $values`, parameters	`array`
`dist_lognormal_quantile()`	Quantile	`float $p`, parameters	`float`
`dist_lognormal_quantile_array()`	Quantile over a list	`array $probabilities`, parameters	`array`
`dist_lognormal_rng()`	Seeded samples	`int $n`, `int $seed`, parameters	`array`

Phit\dist_lognormal_cdf(    float $x,    float $meanlog = 0.0,    float $sdlog = 1.0,): floatPhit\dist_lognormal_cdf_array(    array $values,    float $meanlog = 0.0,    float $sdlog = 1.0,): array

Errors Exception: code 1001 for an invalid parameter, 1003 for a NAN input, or 1006 for a numerical failure.

Sampler errors Exception: code 1001 for invalid input, 1003 for a nonfinite parameter or sample, or 1007 for more than 1,000,000 draws or an unsupported size.

Logistic

Parameter	Type	Default	Description
`$location`	`float`	`0.0`	Location: the mean and median
`$scale`	`float`	`1.0`	Scale, > 0

Function	Operation	Arguments	Returns
`dist_logistic_pdf()`	Density	`float $x`, parameters	`float`
`dist_logistic_pdf_array()`	Density over a list	`array $values`, parameters	`array`
`dist_logistic_logpdf()`	Log density	`float $x`, parameters	`float`
`dist_logistic_logpdf_array()`	Log density over a list	`array $values`, parameters	`array`
`dist_logistic_cdf()`	Lower tail P(X ≤ x)	`float $x`, parameters	`float`
`dist_logistic_cdf_array()`	Lower tail P(X ≤ x) over a list	`array $values`, parameters	`array`
`dist_logistic_logcdf()`	Log lower tail	`float $x`, parameters	`float`
`dist_logistic_logcdf_array()`	Log lower tail over a list	`array $values`, parameters	`array`
`dist_logistic_sf()`	Upper tail P(X > x)	`float $x`, parameters	`float`
`dist_logistic_sf_array()`	Upper tail P(X > x) over a list	`array $values`, parameters	`array`
`dist_logistic_logsf()`	Log upper tail	`float $x`, parameters	`float`
`dist_logistic_logsf_array()`	Log upper tail over a list	`array $values`, parameters	`array`
`dist_logistic_quantile()`	Quantile	`float $p`, parameters	`float`
`dist_logistic_quantile_array()`	Quantile over a list	`array $probabilities`, parameters	`array`
`dist_logistic_rng()`	Seeded samples	`int $n`, `int $seed`, parameters	`array`

Phit\dist_logistic_cdf(    float $x,    float $location = 0.0,    float $scale = 1.0,): floatPhit\dist_logistic_cdf_array(    array $values,    float $location = 0.0,    float $scale = 1.0,): array

Errors Exception: code 1001 for an invalid parameter, 1003 for a NAN input, or 1006 for a numerical failure.

Sampler errors Exception: code 1001 for invalid input, 1003 for a nonfinite parameter or sample, or 1007 for more than 1,000,000 draws or an unsupported size.

Binomial

Parameter	Type	Default	Description
`$size`	`int`	required	Number of trials, an integer in [0, 2^53]
`$prob`	`float`	required	Success probability in [0, 1]

Function	Operation	Arguments	Returns
`dist_binomial_pmf()`	Probability mass	`float $k`, parameters	`float`
`dist_binomial_pmf_array()`	Probability mass over a list	`array $counts`, parameters	`array`
`dist_binomial_logpmf()`	Log probability mass	`float $k`, parameters	`float`
`dist_binomial_logpmf_array()`	Log probability mass over a list	`array $counts`, parameters	`array`
`dist_binomial_cdf()`	Lower tail P(X ≤ x)	`float $k`, parameters	`float`
`dist_binomial_cdf_array()`	Lower tail P(X ≤ x) over a list	`array $counts`, parameters	`array`
`dist_binomial_logcdf()`	Log lower tail	`float $k`, parameters	`float`
`dist_binomial_logcdf_array()`	Log lower tail over a list	`array $counts`, parameters	`array`
`dist_binomial_sf()`	Upper tail P(X > x)	`float $k`, parameters	`float`
`dist_binomial_sf_array()`	Upper tail P(X > x) over a list	`array $counts`, parameters	`array`
`dist_binomial_logsf()`	Log upper tail	`float $k`, parameters	`float`
`dist_binomial_logsf_array()`	Log upper tail over a list	`array $counts`, parameters	`array`
`dist_binomial_quantile()`	Quantile	`float $p`, parameters	`float`
`dist_binomial_quantile_array()`	Quantile over a list	`array $probabilities`, parameters	`array`
`dist_binomial_rng()`	Seeded samples	`int $n`, `int $seed`, parameters	`array`

Phit\dist_binomial_cdf(    float $k,    int $size,    float $prob,): floatPhit\dist_binomial_cdf_array(    array $counts,    int $size,    float $prob,): array

Errors Exception: code 1001 for an invalid parameter, 1003 for a NAN input, or 1006 for a numerical failure.

Sampler errors Exception: code 1001 for invalid input, 1003 for a nonfinite parameter or sample, or 1007 for more than 1,000,000 draws or an unsupported size.

Poisson

Parameter	Type	Default	Description
`$lambda`	`float`	required	Mean, ≥ 0

Function	Operation	Arguments	Returns
`dist_poisson_pmf()`	Probability mass	`float $k`, parameters	`float`
`dist_poisson_pmf_array()`	Probability mass over a list	`array $counts`, parameters	`array`
`dist_poisson_logpmf()`	Log probability mass	`float $k`, parameters	`float`
`dist_poisson_logpmf_array()`	Log probability mass over a list	`array $counts`, parameters	`array`
`dist_poisson_cdf()`	Lower tail P(X ≤ x)	`float $k`, parameters	`float`
`dist_poisson_cdf_array()`	Lower tail P(X ≤ x) over a list	`array $counts`, parameters	`array`
`dist_poisson_logcdf()`	Log lower tail	`float $k`, parameters	`float`
`dist_poisson_logcdf_array()`	Log lower tail over a list	`array $counts`, parameters	`array`
`dist_poisson_sf()`	Upper tail P(X > x)	`float $k`, parameters	`float`
`dist_poisson_sf_array()`	Upper tail P(X > x) over a list	`array $counts`, parameters	`array`
`dist_poisson_logsf()`	Log upper tail	`float $k`, parameters	`float`
`dist_poisson_logsf_array()`	Log upper tail over a list	`array $counts`, parameters	`array`
`dist_poisson_quantile()`	Quantile	`float $p`, parameters	`float`
`dist_poisson_quantile_array()`	Quantile over a list	`array $probabilities`, parameters	`array`
`dist_poisson_rng()`	Seeded samples	`int $n`, `int $seed`, parameters	`array`

Phit\dist_poisson_cdf(    float $k,    float $lambda,): floatPhit\dist_poisson_cdf_array(    array $counts,    float $lambda,): array

Errors Exception: code 1001 for an invalid parameter, 1003 for a NAN input, or 1006 for a numerical failure.

Sampler errors Exception: code 1001 for invalid input, 1003 for a nonfinite parameter or sample, or 1007 for more than 1,000,000 draws or an unsupported size.

Negative binomial (size, prob)

Parameter	Type	Default	Description
`$size`	`float`	required	Dispersion: a positive real, not a trial count
`$prob`	`float`	required	Success probability in (0, 1]

Function	Operation	Arguments	Returns
`dist_nbinom_pmf()`	Probability mass	`float $k`, parameters	`float`
`dist_nbinom_pmf_array()`	Probability mass over a list	`array $counts`, parameters	`array`
`dist_nbinom_logpmf()`	Log probability mass	`float $k`, parameters	`float`
`dist_nbinom_logpmf_array()`	Log probability mass over a list	`array $counts`, parameters	`array`
`dist_nbinom_cdf()`	Lower tail P(X ≤ x)	`float $k`, parameters	`float`
`dist_nbinom_cdf_array()`	Lower tail P(X ≤ x) over a list	`array $counts`, parameters	`array`
`dist_nbinom_logcdf()`	Log lower tail	`float $k`, parameters	`float`
`dist_nbinom_logcdf_array()`	Log lower tail over a list	`array $counts`, parameters	`array`
`dist_nbinom_sf()`	Upper tail P(X > x)	`float $k`, parameters	`float`
`dist_nbinom_sf_array()`	Upper tail P(X > x) over a list	`array $counts`, parameters	`array`
`dist_nbinom_logsf()`	Log upper tail	`float $k`, parameters	`float`
`dist_nbinom_logsf_array()`	Log upper tail over a list	`array $counts`, parameters	`array`
`dist_nbinom_quantile()`	Quantile	`float $p`, parameters	`float`
`dist_nbinom_quantile_array()`	Quantile over a list	`array $probabilities`, parameters	`array`
`dist_nbinom_rng()`	Seeded samples	`int $n`, `int $seed`, parameters	`array`

Phit\dist_nbinom_cdf(    float $k,    float $size,    float $prob,): floatPhit\dist_nbinom_cdf_array(    array $counts,    float $size,    float $prob,): array

Errors Exception: code 1001 for an invalid parameter, 1003 for a NAN input, or 1006 for a numerical failure.

Sampler errors Exception: code 1001 for invalid input, 1003 for a nonfinite parameter or sample, or 1007 for more than 1,000,000 draws or an unsupported size.

Negative binomial (size, mu)

Parameter	Type	Default	Description
`$size`	`float`	required	Dispersion: a positive real, not a trial count
`$mu`	`float`	required	Mean, ≥ 0

Function	Operation	Arguments	Returns
`dist_nbinom_mu_pmf()`	Probability mass	`float $k`, parameters	`float`
`dist_nbinom_mu_pmf_array()`	Probability mass over a list	`array $counts`, parameters	`array`
`dist_nbinom_mu_logpmf()`	Log probability mass	`float $k`, parameters	`float`
`dist_nbinom_mu_logpmf_array()`	Log probability mass over a list	`array $counts`, parameters	`array`
`dist_nbinom_mu_cdf()`	Lower tail P(X ≤ x)	`float $k`, parameters	`float`
`dist_nbinom_mu_cdf_array()`	Lower tail P(X ≤ x) over a list	`array $counts`, parameters	`array`
`dist_nbinom_mu_logcdf()`	Log lower tail	`float $k`, parameters	`float`
`dist_nbinom_mu_logcdf_array()`	Log lower tail over a list	`array $counts`, parameters	`array`
`dist_nbinom_mu_sf()`	Upper tail P(X > x)	`float $k`, parameters	`float`
`dist_nbinom_mu_sf_array()`	Upper tail P(X > x) over a list	`array $counts`, parameters	`array`
`dist_nbinom_mu_logsf()`	Log upper tail	`float $k`, parameters	`float`
`dist_nbinom_mu_logsf_array()`	Log upper tail over a list	`array $counts`, parameters	`array`
`dist_nbinom_mu_quantile()`	Quantile	`float $p`, parameters	`float`
`dist_nbinom_mu_quantile_array()`	Quantile over a list	`array $probabilities`, parameters	`array`
`dist_nbinom_mu_rng()`	Seeded samples	`int $n`, `int $seed`, parameters	`array`

Phit\dist_nbinom_mu_cdf(    float $k,    float $size,    float $mu,): floatPhit\dist_nbinom_mu_cdf_array(    array $counts,    float $size,    float $mu,): array

Errors Exception: code 1001 for an invalid parameter, 1003 for a NAN input, or 1006 for a numerical failure.

Sampler errors Exception: code 1001 for invalid input, 1003 for a nonfinite parameter or sample, or 1007 for more than 1,000,000 draws or an unsupported size.