📂Probability Distribution

Why Normal Distribution

Definition ¹

The probability distribution which has the following probability density function in terms of location $\xi \in \mathbb{R}$, scale $\omega > 0$, and shape $\alpha \in \mathbb{R}$ parameters is called the Skew Normal Distribution. $$ \begin{align*} f(x) =& {{ 2 } \over { \omega }} \phi \left( {{ x - \xi } \over { \omega }} \right) \Phi \left( \alpha {{ x - \xi } \over { \omega }} \right) \\ =& {{ 2 } \over { \omega \sqrt{2 \pi} }} e^{-{{ \left( x - \xi \right)^{2} } \over { 2 \omega^{2} }}} \int_{- \infty}^{\alpha {{ x - \xi } \over { \omega }}} {{ 1 } \over { \sqrt{2 \pi} }} e^{- {{ t^{2} } \over { 2 }}} dt \end{align*} $$ Here, the two functions $\phi, \Phi : \mathbb{R} \to \mathbb{R}$ refer to the probability density function (pdf) and the cumulative distribution function (cdf) of the Standard Normal Distribution, respectively.

Description

The Skew Normal Distribution is, as the name suggests, a generalization of the Normal Distribution with a non-zero skewness $0$. The translation is based on domestic journals.²

Code

Below is Julia code that visualizes how the probability density function of the Skew Normal Distribution changes according to $\alpha$ in an animated way.

using Distributions, Plots

→ = 0:0.1:5
← = reverse(→)
a1 = @animate for α ∈ [→; ←; -→; -←]
    SN = SkewNormal(0, 1, α)
    x = -3:0.01:3
    plot(x, pdf.(SN, x),
    ylim = (0, 0.8),
    lw = 2, color = :black,
    legend = :none, title = "pdf of Skew Normal Distribution with α = $(lpad(α, 4))")
end
gif(a1, string(@__DIR__) * "/skew_normal.gif")