# mot.library_functions.continuous_distributions package¶

## mot.library_functions.continuous_distributions.gamma module¶

class mot.library_functions.continuous_distributions.gamma.gamma_cdf[source]

Calculate the Cumulative Distribution Function of the Gamma function.

This computes: lower_incomplete_gamma(k, x/theta) / gamma(k)

With k the shape parameter, theta the scale parameter, lower_incomplete_gamma the lower incomplete gamma function and gamma the complete gamma function.

Function arguments:

• shape: the shape parameter of the gamma distribution (often denoted $$k$$)
• scale: the scale parameter of the gamma distribution (often denoted $$\theta$$)
class mot.library_functions.continuous_distributions.gamma.gamma_cdf_approx[source]

Approximate the Cumulative Distribution Function of the Gamma function.

This uses the approximation from Revfeim  to compute the cdf for x given the shape and scale parameters.

The approximation returns infinity for values near the tails of the distribution, i.e. where the cdf is near zero or near one.

Function arguments:
• x: the value at which to approximate the cdf
• shape: the shape parameter of the gamma distribution (often denoted $$k$$)
• scale: the scale parameter of the gamma distribution (often denoted $$\theta$$)

References

1. Revfeim, K. J. A. (1991). Approximation for the cumulative and inverse gamma distribution. Statistica Neerlandica, 45(3), 327–331.
class mot.library_functions.continuous_distributions.gamma.gamma_logpdf[source]

Computes the log of the Gamma probability density function using the shape and scale parameterization.

This computes the gamma PDF as:

$\frac{-x}{\theta} + (k-1)\ln(x) - \ln(\Gamma(k)) - k * \ln(\theta)$

With $$x$$ the desired position, $$k$$ the shape and $$\theta$$ the scale.

class mot.library_functions.continuous_distributions.gamma.gamma_pdf[source]

Computes the Gamma probability density function using the shape and scale parameterization.

This computes the gamma PDF as:

${\frac{1}{\Gamma (k)\theta ^{k}}}x^{k-1}e^{-{\frac {x}{\theta }}}$

With $$x$$ the desired position, $$k$$ the shape and $$\theta$$ the scale.

class mot.library_functions.continuous_distributions.gamma.gamma_ppf[source]

Computes the inverse of the cumulative distribution function of the Gamma distribution.

This is the inverse of the Gamma CDF.

class mot.library_functions.continuous_distributions.gamma.gamma_ppf_approx[source]

Approximates the Gamma percentile point function.

This uses the approximation from Revfeim  to compute the ppf for y given the shape and scale parameters.

The approximation is not valid in the tails of the distribution, i.e. where the cdf is near zero or near one.

Function arguments:
• y: the value at which to approximate the ppf
• shape: the shape parameter of the gamma distribution (often denoted $$k$$)
• scale: the scale parameter of the gamma distribution (often denoted $$\theta$$)

References

1. Revfeim, K. J. A. (1991). Approximation for the cumulative and inverse gamma distribution. Statistica Neerlandica, 45(3), 327–331.
class mot.library_functions.continuous_distributions.gamma.igam[source]

Complemented incomplete Gamma integral

Also known as the regularized lower incomplete gamma function. Copied from Scipy (https://github.com/scipy/scipy/blob/master/scipy/special/cephes/igam.c), 05-05-2018:

/*                                                     igam.c
*
*     Incomplete Gamma integral
*
*
*
* SYNOPSIS:
*
* double a, x, y, igam();
*
* y = igam( a, x );
*
* DESCRIPTION:
*
* The function is defined by
*
*                           x
*                            -
*                   1       | |  -t  a-1
*  igam(a,x)  =   -----     |   e   t   dt.
*                  -      | |
*                 | (a)    -
*                           0
*
*
* In this implementation both arguments must be positive.
* The integral is evaluated by either a power series or
* continued fraction expansion, depending on the relative
* values of a and x.
*
* ACCURACY:
*
*                      Relative error:
* arithmetic   domain     # trials      peak         rms
*    IEEE      0,30       200000       3.6e-14     2.9e-15
*    IEEE      0,100      300000       9.9e-14     1.5e-14
*/

class mot.library_functions.continuous_distributions.gamma.igam_fac[source]

Compute x^a * exp(-x) / gamma(a)

Copied from Scipy (https://github.com/scipy/scipy/blob/master/scipy/special/cephes/igam.c), 05-05-2018.

class mot.library_functions.continuous_distributions.gamma.igam_igamc_asymptotic_series[source]

Compute igam/igamc using DLMF 8.12.3/8.12.4.

Copied from Scipy (https://github.com/scipy/scipy/blob/master/scipy/special/cephes/igam.c), 05-05-2018.

The argument func should be 1 when computing for IGAM and 0 when computing for IGAMC.

class mot.library_functions.continuous_distributions.gamma.igam_series[source]

Compute igamc using DLMF 8.11.4

Copied from Scipy (https://github.com/scipy/scipy/blob/master/scipy/special/cephes/igam.c), 05-05-2018.

class mot.library_functions.continuous_distributions.gamma.igamc[source]

Complemented incomplete Gamma integral

Also known as the regularized upper incomplete gamma function. Copied from Scipy (https://github.com/scipy/scipy/blob/master/scipy/special/cephes/igam.c), 05-05-2018:

/*                                                  igamc()
*
*  Complemented incomplete Gamma integral
*
*
*
* SYNOPSIS:
*
* double a, x, y, igamc();
*
* y = igamc( a, x );
*
* DESCRIPTION:
*
* The function is defined by
*
*
*  igamc(a,x)   =   1 - igam(a,x)
*
*                            inf.
*                              -
*                     1       | |  -t  a-1
*               =   -----     |   e   t   dt.
*                    -      | |
*                   | (a)    -
*                             x
*
*
* In this implementation both arguments must be positive.
* The integral is evaluated by either a power series or
* continued fraction expansion, depending on the relative
* values of a and x.
*
* ACCURACY:
*
* Tested at random a, x.
*                a         x                      Relative error:
* arithmetic   domain   domain     # trials      peak         rms
*    IEEE     0.5,100   0,100      200000       1.9e-14     1.7e-15
*    IEEE     0.01,0.5  0,100      200000       1.4e-13     1.6e-15
*/

class mot.library_functions.continuous_distributions.gamma.igamc_continued_fraction[source]

Compute igamc using DLMF 8.9.2.

Copied from Scipy (https://github.com/scipy/scipy/blob/master/scipy/special/cephes/igam.c), 05-05-2018.

class mot.library_functions.continuous_distributions.gamma.igamc_series[source]

Compute igamc using DLMF 8.7.3.

This is related to the series in igam_series but extra care is taken to avoid cancellation.

Copied from Scipy (https://github.com/scipy/scipy/blob/master/scipy/special/cephes/igam.c), 05-05-2018.

class mot.library_functions.continuous_distributions.gamma.igamci[source]

Copied from Scipy (https://github.com/scipy/scipy/blob/master/scipy/special/cephes/igami.c), 05-05-2018.

class mot.library_functions.continuous_distributions.gamma.igami[source]

Copied from Scipy (https://github.com/scipy/scipy/blob/master/scipy/special/cephes/igami.c), 05-05-2018.

## mot.library_functions.continuous_distributions.normal module¶

class mot.library_functions.continuous_distributions.normal.normal_cdf[source]

Compute the Cumulative Distribution Function of the Gaussian distribution.

class mot.library_functions.continuous_distributions.normal.normal_logpdf[source]

Compute the log of the Probability Density Function of the Gaussian distribution.

class mot.library_functions.continuous_distributions.normal.normal_pdf[source]

Compute the Probability Density Function of the Gaussian distribution.

class mot.library_functions.continuous_distributions.normal.normal_ppf[source]

Computes the inverse of the cumulative distribution function of the Gaussian distribution.

This is the inverse of the Gaussian CDF.