Normal (or Gaussian) distribution
is a continuous distribution, defined by a probability density function as in
the following
Where μ is the mean and it is
also the median and mode (for normal distribution mean=median=mode). The
parameter σ is its standard deviation; its variance is therefore σ 2.
A random variable is said to be normal if it has he normal distribution.
The red curve is the standard
normal distribution
Source: Wikipedia
If the normal distribution has the
zero mean and the unity as the standard distribution, it is called a standard
normal distribution.
A variable that is normality
distributed will have the bell shape distribution in which less than one
standard distribution from the mean around 64.2 % of the data as in the dark
blue area in the following shape. That is, the normal variable is that the majority
of the data, 64.2 %, lies around the mean. Around 91.4% of the data lies in
less than 2 times the standard deviation from the mean. In three times the standard 95.6% of the data
are located.
Source:
Wikipedia
The generalization of the univariate normal distribution is
known as the mulltivariate normal distribution or multivariate Gaussian
distribution. It is often used to describe a set of correlated random
variables the values of which are centered around their respective mean values. The Probability Density Function
is given by the following.
Remember:
In a multivariate system: univariate normal
distribution is a necessary but not sufficient condition of Multivariate normal
distribution. In other words, if we have a multivariate normal distribution, the marginal distribution of each dimension is univariate normal.
The normality assumption is a must for hypotheses testing in
parametric statistics.
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