A history of the mathematical theory of probability from the time of Pascal to that of Laplace / by I. Todhunter.

  • Isaac Todhunter
Date:
1865
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Licence: Public Domain Mark

Credit: A history of the mathematical theory of probability from the time of Pascal to that of Laplace / by I. Todhunter. Source: Wellcome Collection.

    590/648 (page 570)
    ; r V{2«r (*--*?)},, e 2s'tf~k?) c]v_ This will be found to agree with Laplace’s page 311. For an example suppose that the function of the facility of error is a constant, say C; then since f /(«) dz = 1, d A we have Thus aC = 1. 2 2 7, __ a y — — y _ p — _ 2’ 3 ’ 12* Therefore the probability that the sum of the errors will lie i 5^ -I SCt > between — rj and — + 77 is 2^6 ay/(sir)J 0 6u2 sas 1006. Laplace next investigates the probability that the sum of the squares of the errors will lie between assigned limits, sup- posing the function of the facility of error to be the same at every observation, and positive and negative errors equally likely. In order to give the result we must first generalise Poisson’s problem. Let </>i [z] denote any function of z: required the probability that 4>i (ei) + $2 (6a) + ... + $« (e«) will lie between the limits c — tj and c + rj. The investigation will differ very slightly from that in Art. 1002. In that Article we have Qi = ffi (*) J b V=i dz; in the present case the exponent of e instead of being <y.XzJ— 1, will be x(f>i [z) J— 1. The required probability will be found to be 2 k V if (l-c+v)2 e 4k2 dv;