Formula for Normal Distribution,Central Limit Theorem:Normal Distribution

In many natural processes, random variation can be done due to a particular probability distribution known as the normal distribution, which can observed commonly in probability distribution. Mathematicians de Moivre and Laplace used this type of distribution in the 1700’s. Before 1800’s, German mathematician and physicist Karl Gauss used this distribution to analyze astronomical data, and it consequently became known as the Gaussian distribution among the scientific community.

The shape of the normal distribution reassembled that of a bell, so it sometimes is referred to as the “bell curve

Formula for Normal Distribution

F(x) = 1/(σ ‘sqrt(2pi)’) e^-1/2((x-µ)/σ)^2

on the domain . The statisticians and mathematicians can uniformly used the term such as “normal distribution” for this distribution, physicists sometimes call it a Gaussian distribution and, because of its curved flaring shape, social scientists refer to it as the “bell curve.” Feller (1968) uses the symbol φ(x)for p(x) in the above equation, but then switches to n(x)in Feller (1971).

De Moivre developed the normal distribution instead of the binomial distribution, and it was subsequently used by Laplace in 1783 to study measurement errors and by Gauss in 1809 in the analysis of astronomical data the normal distribution is implemented in mathematical as normal distribution

Therefore the “standard normal distribution” is given by taking µ=0 and σ^2=1in a general normal distribution. Then the arbitrary normal distribution can be converted into a standard normal distribution by changing variables to z = (x -µ)/σ, so dz = dx / σ, yielding

Central Limit Theorem: Normal Distribution

The Normal distributions has many properties to determination, so random varieties with unknown distributions are often assumed to be normal, especially in physics and astronomy. Although the assumption will be a dangerous, it is often a good approximation due to a surprising result known as the central limit theorem In This theorem the mean of any set of variates with any distribution having a finite mean and variance ends to the normal distribution. Many common attributes such as test scores, height, etc., we can follow roughly this distribution, with few members at the low and high ends and many in the middle.