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Normal Distribution#

The normal (or Gaussian) distribution is a very common continuous probability distribution in probability theory.

Notation \(\mathcal N(μ, σ^2), μ ∈ \R, σ > 0\)
Mean \(μ\)
Variance \(σ^2\)
PDF \(f_X (x) = \frac{1}{\sqrt{2 \pi \sigma^2}} e^{-\frac{(x-\mu)^2}{2 \sigma^2}}\), \(x ∈ \R\)

Probability Density#

\[f_X (x) = \frac{1}{\sqrt{2 \pi \sigma^2}} e^{-\frac{(x-\mu)^2}{2 \sigma^2}}\]

with the mean \(\mu\), the standard deviation \(\sigma\).


  • It is symmetric around the point \(x=\mu\) which is at the same time the mode, the median and the mean of the distribution.
  • The area under the curve and over the x-axis is equal to one.