Normal distribution

The Normal (or Gaussian) distribution is a widely used symmetric continuous probability distribution.

In the following, let \(X\) be a random variable that follows a Normal distribution with mean \(\mu_X\) and standard deviation \(\sigma_X\) , and let \(x\) denote a particular outcome of \(X\).

import fesslix as flx
flx.load_engine()

import numpy as np
import fesslix.plot as flx_plot
import matplotlib.pyplot as plt
%matplotlib inline

Random Number Generator: MT19937 - initialized with rand()=784456830;
Random Number Generator: MT19937 - initialized with 1000 initial calls.

Syntax

property normal

Normal distribution

normal is a distribution type (flx_rv_type) for Random variables in Fesslix.

Parametrization:

Parameters of the distribution can be specified as additional key-value pairs in an object of type flx_rv_config. The following combinations of parameters are accepted:

• mu, sd

• cov, val_1, pr_1

• sd, val_1, pr_1

• val_1, pr_1, val_2, pr_2

The interpretation of the parameters is:

• mu (flxPara): mean value

• sd (flxParaPosNo0): standard deviation

• cov (flxParaPosNo0): coefficient of variation

• val_1 (flxPara): pr_1 quantile

• `pr_1 (flxParaPr): probability that the value of the distribution is smaller or equal than val_1

• val_2 (flxPara): pr_2 quantile

• `pr_2 (flxParaPr): probability that the value of the distribution is smaller or equal than val_2

Example:

rv_1 = flx.rv({'name':'rv_1', 'type':'normal', 'mu':2., 'sd':5. })
rv_2 = flx.rv({'name':'rv_2', 'type':'normal', 'cov':0.1, 'val_1':25., 'pr_1':0.05 })
rv_3 = flx.rv({'name':'rv_3', 'type':'normal', 'sd':2., 'val_1':25., 'pr_1':0.05 })
rv_4 = flx.rv({'name':'rv_4', 'type':'normal', 'val_1':25., 'pr_1':0.05, 'val_2':32., 'pr_2':0.5 })
rv_lst = [ rv_1, rv_2, rv_3, rv_4 ]

Properties

Notation

\(X\sim\mathcal{N}(\mu_X,\sigma_X)\)

Parameters

\(\mu_X\in\mathbb{R}\), \(\sigma_X\in(0,\infty)\)

Support

\(x\in\mathbb{R}\)

Mean

\(\mu_X\)

Standard deviation

\(\sigma_X\)

Median

\(\mu_X\)

Mode

\(\mu_X\)

Skewness

zero

Excess kurtosis

zero

Entropy

\(\frac{1}{2}\ln\left(2\pi e {\sigma_X}^2\right)\)

PDF

The PDF \(f_X(x)\) of the Normal distribution is commonly expressed in terms of the PDF of the Standard Normal distribution \(\varphi(\cdot)\):

\[f_X(x) = \frac{1}{\sigma_X} \cdot\varphi\left(\frac{x-\mu_X}{\sigma_X}\right)\]

The PDF of a Normal distribution can also be expressed as:

\[f_X(x) = c \cdot\exp\left(-\lambda_1 x - \lambda_2 x^2\right)\]

where the mean, standard deviation and scaling constant are:

\[\mu = -\frac{\lambda_1}{2\lambda_2}\]

\[\lambda_1 = - \frac{\mu}{\sigma^2}\]

\[\sigma = \frac{1}{\sqrt{2\lambda_2}}\]

\[\lambda_2 = \frac{1}{2 \sigma^2}\]

\[c = \sqrt{\frac{\lambda_2}{\pi}} \cdot\exp\left(-\frac{\lambda_1^2}{4\lambda_2}\right)\]

From the equations above, it follows that \(\lambda_2\) must be positive.

fig, ax = plt.subplots(figsize=(10, 4))

for rv in rv_lst:
    flx_plot.draw_pdf(ax, rv, config_dict={'label':rv.get_name()}) 

ax.set_ylim([0., None])
plt.xlabel(r"$x$")
plt.ylabel(r"$f_X(x)$")
plt.legend()
plt.show()

CDF

The CDF \(F_X(x)\) of the Normal distribution is typically expressed in terms of the CDF of the Standard Normal distribution \(\Phi(\cdot)\):

\[F_X(x) = \Phi\left(\frac{x-\mu_X}{\sigma_X}\right)\]

fig, ax = plt.subplots(figsize=(10, 4))

for rv in rv_lst:
    flx_plot.draw_cdf(ax, rv, config_dict={'label':rv.get_name()}) 
    
ax.set_ylim([0., 1.])
plt.xlabel(r"$x$")
plt.ylabel(r"$F_X(x)$")
plt.legend()
plt.show()

Quantile function

The quantile function of the Normal distribution is:

\[F_X^{-1}(p) = \mu_X + \sigma_X \cdot\Phi^{-1}(p) \;, \quad p\in(0,1)\]

where \(\Phi^{-1}(\cdot)\) is the quantile function of the standard Normal distribution.

Standardizing Normal random variables

The Normal random variable \(X\) can be transformed to a Standard Normal distribution \(U\) through:

\[U = \frac{X-\mu_X}{\sigma_X}\]

Conversely, a Normal random variable \(X\) with mean \(\mu_X\) and standard deviation \(\sigma_X\) can be generated from a standard Normal variable as:

\[X = \mu_X + \sigma_X \cdot U\]