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

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 $${\mathcal N}_{\gamma}(t)=\int_t^{\infty}x^{\gamma}e^{-x^2/2}dx, t\geq 0$$.

MSC:

 6e+06 Probability distributions: general theory 6e+16 Inequalities; stochastic orderings