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Bayes estimator of the Maxwell’s velocity distribution function. (English) Zbl 0718.62068
Summary: Suppose there is available a random sample of observed values $$(x_ 1,x_ 2,...,x_ n)$$ from a Maxwell velocity distribution, for a randomly chosen molecule of a gas, specified by the pdf: $f_{\theta}(x)=4\pi^{-1/2}\theta^{-3/2}x^ 2e^{-x^ 2/\theta}\quad (0\leq x<\infty;\quad \theta >0).$ Also suppose there is some prior knowledge about the parameter $$\theta$$ summarized in a natural conjugate Bayesian density of the inverted gamma type $g(\theta) \propto \theta^{-(\nu +1)}e^{-\tau /\theta}\quad (0<\theta <\infty;\quad \tau,\nu >0).$ Then, under the assumption of squared error loss function, the Bayes estimator of the average speed of the molecule is worked out, and also the Bayes estimator of the Maxwell d.f. is obtained in terms of the Gauss hypergeometric function. From the estimator of the d.f., one can estimate the probability that the speed of the molecule lies between two prescribed values.

##### MSC:
 62F15 Bayesian inference 62P99 Applications of statistics