Two-sided estimates of Gamma distribution. (Ukrainian, English) Zbl 1027.60006

Vozovitov, O. O.

Teor. Jmovirn. Mat. Stat. 65, 20-24 (2001); translation in Theory Probab. Math. Stat. 65, 21-26 (2002).

The author proposes the lower and the upper bounds for the integral \({\mathcal T}_r(t)=\int_t^{\infty}x^re^{-x}dx, t\geq 0,r\geq 0\). This integral can be represented in the form \({\mathcal T}_r(t)=A{\mathcal T}_{\alpha}(t)+B(t),t\geq 0,\alpha\in[0,1)\), where \(\alpha=r-[r], A=r(r-1)\cdots(\alpha+1)\), \(B(t)=e^{-t}(t^r+rt^{r-1}+\cdots+At^{\alpha})\). It is proved that \(\Phi(4)\leq{\mathcal T}_{\alpha}(t)\leq\Phi(c)\), where \(\Phi(c)= {(4\alpha t^{\alpha}e^{-t})}/( {2t-2\alpha+1+\sqrt{(2t-2\alpha+1)+2ct}})\), when \(t\geq t_0\) for a special \(t_0\). Similar bounds are proposed for the integral \({\mathcal B}_{\gamma}(t)=\int_t^{\infty}x^{\gamma}dF(x), t\geq 0,\) where \(F(x)=1-e^{-\theta x^{\beta}}\), \(x\geq 0\), is the Weibull distribution function, and for the integral \({\mathcal N}_{\gamma}(t)=\int_t^{\infty}x^{\gamma}e^{-x^2/2}dx, t\geq 0\).

Reviewer: Mikhail Moklyachuk (Kyïv)

MSC:

60E05	Probability distributions: general theory
60E15	Inequalities; stochastic orderings

Keywords:

integral; lower bound; upper bound; Weibull distribution