kniga 7 - Probability and Statistics 1 - Sheynin, Oscar

Γ(n / 2)Γ [( n −1) / 2]k = (1/2), k 1 = (1/2), (104)Γ [( n + 1) / 2]Γ(n / 2)i.e., of choosing σ and σ1(see above) in accord with these magnitudes. However, after thatit is natural to take as an approximation to hΓ [( n + 1) / 2]h = (1/S)Γ(n / 2), h1= (1/S 1 )Γ(n / 2)Γ [( n −1) / 2](102bis)in Problems 2 and 3 respectively.If the lack of systematic error is demanded in the determination of 2 (of the variance), andif the approximation to 2 is being determined in the form σ 2 = qS 2 (Problem 2) and σ 2 1=q 1 S 2 1 (Problem 3), then we have to assumeq = 1 – n, q 1 = 1/(n – 1), σ 2 =S 2 /n, σ 2 1= S 2 1 /(n – 1). (105)To achieve conformity with formula (103), it is then natural to assume the approximationto h ash =n / 2S, h1= ( n − 1) / 2S1(98ter)in Problems 2 and 3 respectively. The last approximations are the most generally accepted inmodern mathematical statistics. We have applied them in §4, where, however, the choicebetween (98), (98bis) and (98ter) was not essential since the limit theorems of §4 persistedanyway.From the practitioner’s viewpoint, the differences between the three formulas are notessential when only once determining the measure of precision h by means of (1). Indeed, thedifferences between these approximations have order h/n, and the order of their deviationsfrom the true value 20 , h, is higher, h/n. Therefore, since we are only able to determine h towithin deviations of order h/n, we may almost equally well apply any of theseapproximations which differ one from another to within magnitudes of the order h/n.The matter is quite different if a large number of various measures of precision (forexample, corresponding to various conditions of gunfire) has to be determined, each timeonly by a small number of observations. In this case, the absence of a systematic error insome magnitude, calculated by issuing from the approximate value of the measure ofprecision, can become very essential. Depending on whether this magnitude is, for example,h, or 2 , the approximate values of h should be determined by formulas (98), (98bis) or(98ter) respectively.In particular, from the point of view of an artillery man, according to the opinion of Prof.Gelvikh [10; 11] 21 , it is most essential to determine without systematic error the expectedexpenditure of shells required for hitting a target. In the most typical cases (two-dimensionalscattering and a small target as compared with the scattering) this expected expenditure,according to him, is proportional to the product (1) (2) of the mean square deviations in thetwo directions. Suppose that we estimate (1) and (2) by their approximations σ (1) and σ (2)derived from observations (x 1 (1) , x 2 (1) , …, x n (1) ) and (x 1 (2) , x 2 (2) , …, x m (2) ) respectively. If thex i (1) are independent of x j (2) , then, for any (1) , (2) , a (1) and a (2) (where the last two magnitudesare the centers of scattering for x i (1) and x j (2) respectively), we haveE(σ (1) σ (2) |a (1) ; a (2) ; (1) ; (2) ) = E(σ (1) |a (1) ; (1) ) E(σ (2) |a (2) ; (2) )