In mathematics a Gaussian Distribution Function is a function of the form:

Distribution |
Functional Form |
Mean |
Standard Deviation |

Ganussian |
f |
a |
${\sigma}_{\begin{array}{l}\\ \end{array}}$ |

**The Gaussian distribution is one of the most
commonly used probability distribution for applications. If the number of
events is very large, then the Gaussian distribution function may be used to
describe physical events. The Gaussian distribution is a continuous function
which approximates the exact binomial distribution of events **:

$P(y)=\frac{1}{\sigma \sqrt{2\pi}}e{\frac{-(y-\mu )}{2{\sigma}^{2}}}^{2}$

$\mu $:mean
of distribution $$ ${\sigma}^{2}$:variance
of the distribution y is a continuous variable $(-\infty \le y\le \infty )$

Probability (P) of y being in the range [a, b] is given by an integral

**$P(y<a<b)=\frac{1}{\sigma \sqrt{2\pi}}{\displaystyle \underset{a}{\overset{b}{\int}}{e}^{\frac{-{(y-\mu )}^{2}}{2{\sigma}^{2}}}}{d}_{y}$**

Total area under the curve is normalized to one.

The probability integral: $P(-\infty <y<\infty )=\frac{1}{\sigma \sqrt{2\pi}}{\displaystyle \underset{-\infty}{\overset{\infty}{\int}}{e}^{\frac{-{(y-\mu )}^{2}}{2{\sigma}^{2}}}}{d}_{y}=1$

**The
Gaussian distribution is also commonly called the "normal distribution"
and is often described as a "bell-shaped curve".**

The Diagram shows Normalized Gaussian curves with expected value ${\mu}_{}$ and variance ${\sigma}_{}^{2}$ The corresponding parameters are a= $\frac{1}{(\sigma \sqrt{(2\pi ))}}$, b= $\mu $, c= $\sigma $. $$$$

The normal distribution with mean 0 and standard deviation 1 N(0;1)is called the standard normal distribution.

A random variable with the standard normal distribution is called a standard normal random variable and is usually denoted by Z.

The cumulative probability distribution of the standard normal distribution $P(Z\le z)$

has been tabulated and is used to calculate probabilities for any normal random variable.

$P(Z\le {z}_{0})$ gives the area under the curve to the left of ${z}_{0}$

$P({z}_{0}\le Z\le {z}_{1})=P(Z\le {z}_{1})-P(Z\le {z}_{0})$

The distribution is symmetric $P(Z\le {z}_{0})=P(Z\ge -{z}_{0})$

Example: Look at the graph below and suppose Z is a standard random variable. Calculate:

$\begin{array}{l}P(Z\le 1.1)\\ P(Z>0.8)\\ P(Z\le -1.52)\\ P(0.4\le Z\le 1.32)\\ P(-0.2\le Z\le 0.34)\end{array}$

ONLINE QUIZ ABOUT GAUSSIAN DISTRIBUTION

Click here to go to next page: Real life applications of Gaussian Distribution