1.2 Notation
| \(X_1, X_2, \cdots , X_n\) | Unordered variates | |
| \(x_1, x_2, \cdots , x_n\) | un ordered observations | |
| \(X_{(1)} \leq X_{(2)} \leq \cdots \leq X_{(n)}\) | ordered variates. | |
| \(x_{(1)}\leq x_{(2)}\leq \cdots \leq x_{(n)}\) | ordered observations | |
| \(P(X\leq x)\) | cumulative distribution function | |
| \(P_n(x) \) | \(\dfrac {\# \,\,\text {of}\,\, x_i's\, \text {less than or equal to}\, \, x}{n}\) | |
| empirical commulative distribution function | ||
| \(\Im _P\) | population quantile of order \(P\), given by \(P(\Im _p) = P\) | |
| \(X([np]+1) \) | Sample quantile of order \(p\) where \([np]\) denotes | |
| the largest integer less than or equal to \(np\). | ||
| \(B(a,b) = \displaystyle {\int ^1_0 t^{a - 1}\, (1 - t)^{b - 1}\, dt}\) | Beta function | |
| \(B(a,b) = \dfrac {\Gamma (a)\, \Gamma (b)}{\Gamma (a + b)}\) | ||
| \(\displaystyle {B_p(a,b) = \int ^p_{0}t^{a - 1}\, (1 - t)^{b - 1}\, dt}\) | incomplete Beta integral \(p\in (0,1)\) | |
| \(\Gamma (a) = \displaystyle {\int ^{\infty }_0t^{a - 1}\, e^{-t}\, dt}\) | gamma function | |
| \(\Gamma _{k}(a) = \displaystyle {\int _0^{k}t^{a - 1}\, e^{-t}\, dt}\) | incomplete gamma integral | |
| \(\Gamma (a) = (a - 1)!\) | If \(a\) is a positive integer |
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