# Normal distribution A normal distribution is a type of continuous probability distribution for a real-valued random variable. A random variable with a Gaussian distribution is said to be normally distributed, and is called a normal deviate. Normal distributions are important in statistics and are often used in the natural and social sciences to represent random variables whose distributions are not known.

### About Normal distribution in brief In probability theory, a normal distribution is a type of continuous probability distribution for a real-valued random variable. A random variable with a Gaussian distribution is said to be normally distributed, and is called a normal deviate. Normal distributions are important in statistics and are often used in the natural and social sciences to represent random variables whose distributions are not known. Their importance is partly due to the central limit theorem, which states that, under some conditions, the average of many samples of a random variable is itself a normal distribution. Many results and methods, such as propagation of uncertainty and least squares parameter fitting, can be derived analytically in explicit form when the relevant variables are normally distributed. A normal distribution is sometimes informally called a bell curve, but many other distributions are bell-shaped. Every normal Distribution is a version of the standard normal distribution, whose domain has been stretched by a factor σ{displaystyle sigma } and then translated by μ {displaystyle mu } : The probability density must be scaled by 1σ{display style 1sigma } so that the integral is still 1. The form of the Greek letter phi is often denoted with the Greek word phi, φi, or N(μ, N) The standard Gaussian  distribution is often referred to as N(σ2, N), or Nμ, or the N-normal distribution.

Some authors advocate using the precision of the parameter tau instead of the variance of the distribution, defining the width of the deviation as the standard deviation or σ(tau) instead of σ (sigma) or N (n) The standard normal distribution is also called the standardized normal distribution and is quite often quite often used quite quite often to define the standard regular distributive form of the probability density function. It is a special case when μ=0, and σ=1, and it is described by this probability density function: Here, the factor 12π(x) ensures that the total area under the curve φ(x){displaystyle varphi } is equal to one. The factor 12{display Style 12} in the exponent ensures that the distribution has unit variance, and therefore also unit standard deviation. This function is symmetric around x=0,. where it attains its maximum value 12 ω (x) and has inflection points at x=+1, x=−1, x= +1,  x=-1 and x= −1. If Z is a standard normal deviates, then X=σZ+μ{ displaystyle X=sigma Z+mu } will have a normal  distribution with expected value   and standard deviation   of 1. If X is a normal Deviate, then the variate form of X is called the standard normal distribution and is also known as the X-deviate.