. The Bell System technical journal . It was found that this observeddistribution could be closely approximated by a normal law fo{X)for which the standard deviation cto was A/o-r+c^. This experimentsuggests a general theorem which will be demonstrated analyticalhin a succeeding paragraph. The theorem is: When the true distri-bution friX) and the law of error fE{x) are both normal (hence ex-pressible in form indicated by Equation (1)) with root mean squareor standard deviations <^t and (^e respectively, the most probable ob-served distribution will be normal in form with a standard deviatio
. The Bell System technical journal . It was found that this observeddistribution could be closely approximated by a normal law fo{X)for which the standard deviation cto was A/o-r+c^. This experimentsuggests a general theorem which will be demonstrated analyticalhin a succeeding paragraph. The theorem is: When the true distri-bution friX) and the law of error fE{x) are both normal (hence ex-pressible in form indicated by Equation (1)) with root mean squareor standard deviations <^t and (^e respectively, the most probable ob-served distribution will be normal in form with a standard deviationaro= y/ crx-\- (Je- The observed distribution in Fig. 1 is asymmetrical and hence not 16 BELL SYSTEM TECHNICAL JOURNAL normal as il should be if fr{X) and fE{x) were both normal. Wemust therefore, try some other function ior friX). Of course, experiments might be performed for other t\pes of trueand error distributions, but in all such cases the results, as in theillustration just considered, would be subject to errors of CHARACTERISTIC Fig. 4—Experimental results shpwing efifects of errors of measurement. Normal curvefitted to observed points, when the true distribution and the law of error are both normal Hence we shall proceed at once to the analytical treatment of theproblem. Assuming the law of error to be normal, we see that the fractionfE{x)dx of the number of objects having magnitudes between X+xand X-hx-\-dx will be measured with an error between —x and —x — dxand hence will be observed as of magnitude X (Fig. 5). Thus /oo 1 _ *^ MX^x) -y=e-^.dx. (2) For the particular case treated in a previous paragraph where boththe true distribution friX) and the law of error //i(.T) are normal,we may write Equation (2) in the form ^ /»oo (A+-V)- -v^ fo{X)dX = ^ / e 2,t|. e -a^ dX dx (y-f(Ji^Z (3) where o^r and <^e are the root mean square or standard deviations of CORRliCI lOX Ul IOR MI:. 17 the true and error clislribuLujiis re
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