Normal Distribution

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Regarding Statistical Distributions, there are many different types. Among them, the most important one is the Normal (Gaussian) Distribution. The corresponding PDF (Probabilistic Density Function) is given as follows:

f(x)=\frac{1}{\sqrt{2\pi\sigma^2}}\exp \Bigl( -\frac{(x-\mu)^2}{2\sigma^2}\Bigr)

 

Here, \mu indicates Average (mean), and \sigma denotes Standard Deviation.

The below Figure shows the Probabilistic Density Function of Normal Distribution, in case that Average is \mu=170 and Standard Deviation is \sigma=6 . This example may correspond to the body male height for one population.

正規分布0001

As you can see from Figure, the peak of the distribution is around \mu=170 ㎝. Roughly speaking, the width of the hill at the half-height corresponds to the Standard Deviation.

As a check, we can compute the expectation E and Variance V, which give the following results:

E=\int_{-\infty}^{\infty}x p(x) dx=\mu

 

and

V=\int_{-\infty}^{\infty}(x-\mu)^2 p(x)dx=\sigma^2 .

 

＊We also explain this page in the following video.

 

 

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