Table 1 Approximation of some classical probability distribution
Distribution | Abscissa | Probability | Weight | Polynomial |
N(μ, σ 2) | \( \mu +\sigma\;{x}_j^{(GH)} \) | \( \frac{1}{\sqrt{\pi }}{w}_j^{(GH)} \) | exp(−x 2), − ∞ < x < ∞ | GH |
gamma (t, α) | \( \frac{1}{\alpha}\;{x}_j^{(GLa)} \) | \( \frac{1}{\varGamma t}{w}_j^{(GLa)} \) | \( {x}^{t-{1}^{\prime }} \exp \left(-x\right),\;0<x<\infty \) | GLa |
beta (α, β) | \( \frac{\left(1+{x}_j^{(GJ)}\right)}{2} \) | \( \frac{t^{1-\alpha -\beta}\;{w}_j^{(GJ)}}{beta\;\left(\alpha, \beta \right)} \) | \( \frac{{\left(1-x\right)}^{\alpha -1}}{{\left(1+x\right)}^{1-\beta }},-1<x<1 \) | GJ |
uniforn(a, b) | \( a+\frac{b-a}{2\left(1+{x}_j^{(GLe)}\right)} \) | \( \frac{1}{2}{w}_j^{(GLe)} \) | 1, − 1 < x < 1 | GLe |