The Central Limit Theorem is one of the greatest results in probability theory because it says that the sum of a big number of variables has approximately a normal distribution.

We define a set of random variables iid *X1, X2, X3, … , Xn* with mean *μ* and variance *σ²*. Then if n is very big, the sum

*X1 + X2 + X3 + … + Xn*

is approximately normal with mean *nμ* and variance *nσ²*. If we also normalize the sum we can say that

(*X1 + X2 + X3 + … + Xn – nμ)/(σ√n)*

is approximately a normal standard, so to a normal with mean 0 and variance 1.

__Look for the most popular distributions in statistics__

Distribution can be divided into two categories: discrete and continuous distributions.

The most popular discrete distributions are:

1) Boolean (Bernoulli) which takes value 1 with probability *p* and value 0 with probability *q* = 1 − *p*.

2) Binomial which describes the number of successes in a series of independent Yes/No experiments all with the same probability of success

3) Poisson which describes a very large number of individually unlikely events that happen in a certain time interval

4) Hypergeometric which describes the number of successes in the first *m* of a series of *n* consecutive Yes/No experiments, if the total number of successes is known. This distribution arises when there is no replacement.

The most popular continuous distributions are:

1) Normal (or Gaussian) often used in the natural and social sciences to represent real-valued random variables

2) Chi-squared which is the sum of the squares of *n* independent Gaussian random variables. It is a special case of the Gamma distribution

3) Gamma which describes the time until *n* consecutive rare random events occur in a process with no memory

4) Beta a family of two-parameter distributions with one mode, of which the uniform distribution is a special case, and which is useful in estimating success probabilities

5) T-Student useful for estimating unknown means of Gaussian populations

6) F-Distribution (Fisher) is a continuous probability distribution that arises frequently as the null distribution of a test statistic, most notably in the analysis of variance

7) Weibull of which the exponential distribution is a special case is used to model the lifetime of technical devices and is used to describe the particle size distribution of particles generated by grinding, milling and crushing operations

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