Normal Distribution

Table of contents

Density Function

The density function of the Normal (Gaussian) distribution:

\[f(x; \mu, \sigma) = \frac{1}{\sqrt{2 \pi} \sigma} \exp \left( - \frac{(x-\mu)^2}{2 \sigma^2} \right)\]

Methods for scalar input, as well as for list input, are listed below.

Scalar Input

pystats.dnorm(x: float, mean: float = 0.0, sd: float = 1.0, log: bool = False) → float

Density function of the Normal distribution.

Example

>>> pystats.dnorm(1.645, 0.0, 1.0)
0.10311081109198143
Parameters

  • x (float) – A real-valued input.

  • mean (float) – The mean parameter, a real-valued input.

  • sd (float) – The standard deviation parameter, a real-valued input.

  • log (bool) – Return the log-density or the true form.

Returns

The density function evaluated at x.

List Input

pystats.dnorm(x: List[float], mean: float = 0.0, sd: float = 1.0, log: bool = False) → List[float]

Density function of the Normal distribution.

Example

>>> pystats.dnorm([-1.0, 0.0, 2.0], 0.0, 1.0)
[0.24197072451914342, 0.39894228040143265, 0.05399096651318805]
Parameters

  • x (List[float]) – A standard list input.

  • mean (float) – The mean parameter, a real-valued input.

  • sd (float) – The standard deviation parameter, a real-valued input.

  • log (bool) – Return the log-density or the true form.

Returns

A list of density values corresponding to the elements of x.

Cumulative Distribution Function

The cumulative distribution function (CDF) of the Normal (Gaussian) distribution:

\[F(x; \mu, \sigma) = \int_{-\infty}^x f(z; \mu, \sigma) dz = \frac{1}{2} \times \left( 1 + \text{erf} \left( \frac{x - \mu}{\sqrt{2} \sigma} \right) \right)\]

where \(\text{erf}(\cdot)\) denotes the Gaussian error function.

Methods for scalar input, as well as for list input, are listed below.

Scalar Input

pystats.pnorm(p: float, mean: float = 0.0, sd: float = 1.0, log: bool = False) → float

Distribution function of the Normal distribution.

Example

>>> pystats.pnorm(1.645, 0.0, 1.0)
0.9500150944608786
Parameters

  • p (float) – A real-valued input.

  • mean (float) – The mean parameter, a real-valued input.

  • sd (float) – The standard deviation parameter, a real-valued input.

  • log (bool) – Return the log-density or the true form.

Returns

The cumulative distribution function evaluated at p.

List Input

pystats.pnorm(p: List[float], mean: float = 0.0, sd: float = 1.0, log: bool = False) → List[float]

Distribution function of the Normal distribution.

Example

>>> pystats.pnorm([-1.0, 0.0, 2.0], 0.0, 1.0)
[0.15865525393145705, 0.5, 0.9772498680518208]
Parameters

  • p (List[float]) – A standard list input.

  • mean (float) – The mean parameter, a real-valued input.

  • sd (float) – The standard deviation parameter, a real-valued input.

  • log (bool) – Return the log-density or the true form.

Returns

A list of CDF values corresponding to the elements of p.

Quantile Function

The quantile function of the log-Normal distribution:

\[q(p; \mu, \sigma) = \mu + \sqrt{2} \sigma \times \text{erf}^{-1} \left( 2 p - 1 \right)\]

where \(\text{erf}^{-1}(\cdot)\) denotes the inverse Gaussian error function.

Methods for scalar input, as well as for list input, are listed below.

Scalar Input

pystats.qnorm(q: float, mean: float = 0.0, sd: float = 1.0) → float

Quantile function of the Normal distribution.

Example

>>> pystats.qnorm(0.95, 0.0, 1.0)
1.6448536269514706
Parameters

  • q (float) – A real-valued input.

  • mean (float) – The mean parameter, a real-valued input.

  • sd (float) – The standard deviation parameter, a real-valued input.

Returns

The quantile function evaluated at q.

List Input

pystats.qnorm(q: List[float], mean: float = 0.0, sd: float = 1.0) → List[float]

Quantile function of the Normal distribution.

Example

>>> pystats.qnorm([0.3, 0.5, 0.95], 0.0, 1.0)
[-0.5244005127080409, 0.0, 1.6448536269514706]
Parameters

  • q (List[float]) – A standard list input.

  • mean (float) – The mean parameter, a real-valued input.

  • sd (float) – The standard deviation parameter, a real-valued input.

Returns

A list of quantiles values corresponding to the elements of q.

Random Sampling

Random sampling for the Normal distribution is achieved via the normal_distribution class from the C++ standard library, imported from <random>.

Scalar Output

pystats.rnorm(mean: float = 0.0, sd: float = 1.0) → float

Random sampling function for the Normal distribution.

Example

>>> pystats.rnorm(0.0, 1.0)
0.2772437359624193
Parameters

  • mean (float) – The mean parameter, a real-valued input.

  • sd (float) – The standard deviation parameter, a real-valued input.

Returns

A pseudo-random draw from the Normal distribution.

List Output

pystats.rnorm(n: int, mean: float = 0.0, sd: float = 1.0) → List[float]

Random sampling function for the Normal distribution.

Example

>>> pystats.rnorm(3, 0.0, 1.0)
[-0.04796221296447198, -2.165514623383353, 0.7219224921308472]
Parameters

  • n (int) – The number of output values.

  • mean (float) – The mean parameter, a real-valued input.

  • sd (float) – The standard deviation parameter, a real-valued input.

Returns

A list of pseudo-random draws from the Normal distribution.