Normal Distribution#
The normal (or Gaussian) distribution is a very common continuous probability distribution in probability theory.
| Parameters | |
|---|---|
| Notation | \(\mathcal N(μ, σ^2), μ ∈ \R, σ > 0\) |
| Mean | \(μ\) |
| Variance | \(σ^2\) |
| \(f_X (x) = \frac{1}{\sqrt{2 \pi \sigma^2}} e^{-\frac{(x-\mu)^2}{2 \sigma^2}}\), \(x ∈ \R\) |
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 \(\sigma\).
Properties#
- 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.