User:Abuowda Issa/sandbox/Normal Distribution

'''Definition of Normal Distribution

In probability theory, the normal (or Gaussian or Gauss or Laplace–Gauss) distribution is a very common continuous probability distribution. Normal distributions are important in statistics and are often used in the natural and social sciences to represent real-valued random variables whose distributions are not known. A random variable with a Gaussian distribution is said to be normally distributed and is called a normal deviate. [1]

The normal distribution is sometimes informally called the bell curve. The probability density of the normal distribution is

$ {\displaystyle f(x\mid \mu ,\sigma ^{2})={\frac {1}{\sqrt {2\pi \sigma ^{2}}}}e^{-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}}} $ where
 * undefined

$$ {\displaystyle \mu } $$ is the mean or expectation of the distribution (and also its median and mode) $$ {\displaystyle \sigma }$$ is the standard deviation $$ {\displaystyle \sigma ^{2}} $$ is the variance

The simplest case of a normal distribution is known as the standard normal distribution. This is a special case when $$ {\displaystyle \mu =0}$$ and $$ {\displaystyle \sigma =1}$$, and it is described by this probability density function:

The normal distribution with density $$ {\displaystyle f(x)} ( $$ mean $$ {\displaystyle \mu }  $$  and standard deviation $$  {\displaystyle \sigma >0})  $$ has the following properties:[2]
 * It is symmetric around the point $$ {\displaystyle x=\mu ,} $$ which is at the same time the mode, the median and the mean of the distribution.
 * The area under the curve and over the $$ {\displaystyle x}$$-axis is unity (i.e. equal to one).
 * The probability of a normally distributed variable $$ {\displaystyle X}$$ with known $${\displaystyle \mu }$$ and $$ {\displaystyle \sigma }$$ is in a particular set, can be transferred using the fraction $$ {\displaystyle Z=(X-\mu )/\sigma }$$ into a standard normal distribution.
 * If $$ X_1 $$ and $$ X_2 $$ are two independent standard normal random variables with mean 0 and variance 1, then their sum and difference is distributed normally with mean zero and variance two: $$ X_1 $$ ± $$ X_2 $$ ~ N(0, 2).

Remark: Many distributions can be transferred into normal distribution under certain conditions. For example, if $$ x $$ is a random variable with binomial distribution $$ B(n, p) $$, then for sufficiently large n, the following random variable has a standard normal distribution where $$ \mu= np $$ and $$ \sigma^{2} = np(1-p) $$.

Example: The average IQ of the Hungarian people is 98 and the standard deviation is 8, what is the probability of having people with IQ less than 80 ? Solution: $$ P(X < 80) = P \big( Z < (X- \mu) / \sigma \big) = P ( Z < (80-98)/8 ) = P( Z< -1.2 ) = 0.11507 $$. It means that there is $$ 11.5 % $$ of the Hungarian population with IQ less than 80. References: [1], [2] https://en.wikipedia.org/wiki/Normal_distribution