Normal distribution

Plots the CDF and PDF graphs for normal distribution with given mean and variance.

Michele

Created: 2015-11-28 20:40:11, Last updated: 2020-11-27 19:56:31

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Normal distribution takes a unique role in the probability theory. This is the most common continuous probability distribution, commonly used for random values representation of unknown distribution law.

Probability density function

Normal distribution probability density function is the Gauss function:
$f(x) = \tfrac{1}{\sigma\sqrt{2\pi}}\; e^{ -\frac{(x-\mu)^2}{2\sigma^2} }$

where μ — mean,
σ — standard deviation,
σ ² — variance,
Median and mode of Normal distribution equal to mean μ.

The calculator below gives probability density function value and cumulative distribution function value for the given x, mean, and variance:

Normal distribution

Digits after the decimal point: 5
Probability density function value

Cumulative distribution function value

PDF Graph
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CDF Graph
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Cumulative distribution function

Normal distribution cumulative distribution function has the following formula:
$\frac12\left[1 + \operatorname{erf}\left( \frac{x-\mu}{\sigma\sqrt{2}}\right)\right]$
where, erf(x) - error function, given as:
$\operatorname{erf}(x) = \frac{2}{\sqrt{\pi}}\int\limits_0^x e^{-t^2}\,\mathrm dt$

Quantile function

Normal distribution quantile function (inverse CDF) given as inverse error function:

$F^{-1}(p) = \mu + \sigma\sqrt2\,\operatorname{erf}^{-1}(2p - 1)$
p lays in the range [0,1]

Standard normal distribution quantile function (σ =1, μ=0) equates like this:
$\Phi^{-1}(p)\; =\; \sqrt2\;\operatorname{erf}^{-1}(2p - 1)$
This function is called the probit function.

The calculator below gives quantile value by probability for the specified through mean and variance normal distribution( set variance=1 and mean=0 for probit function).

Normal Distribution Quantile function

Digits after the decimal point: 2
Quantile

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