By Pierre Bremaud

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**Additional resources for Markov Chains: Gibbs Fields, Monte Carlo Simulation, and Queues**

**Example text**

43 OBSERVATION AND EXPEOTA'l'ION number of degrees of freedom available. It was first investigated by Student (11)08) and has been tabulated by R. A. Fisher. It is given in Table 7, and also in the Appendix. The relative areas listed correspond to those of Table 4. The bottom line of Table 7, in fact, contains the same entries as Table 4 (except that the latter have been rounded off to four decimals) for the degrees of freedom upon which 8 is based are here taken as infinity, and, consequently, s, for these values of t, is fj exactly.

0)2 = 288 and since it is based upon a single degree of freedom, this is also the estimate of variance among the random sampling units of block 1, though not yet adjusted for the finite population sampled. When an estimate of variance is based upon just two random sampling unit observations, a short-cut method of calculation is to be pre_ ferred. Let Xl and X2 be two such observations from an infinite popula1 tion of x. 2) is r r which may be written [ r me~Lll ~ (Xl-X2) + [ ; (X2- Xl) or, since these two terms are identical, as ~ (:l:1-X2)2 on one degree of freedom.

But if the sample marbles are not replaced during the course of the n drawings, the random sample is from the finite population of 100 numbers. In the latter case n is, of course, less than 100. Finite populations are the rule in most of the sampling problems with which forestry is concerned. Timber cruising is a sampling job on a finite area of timber stand or forest. The estimate of natural reproduction on logged-over areas, and the evaluation of forage in the meadow, are everyday problems of the forester in sampling finite populations.