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.

    602/648 (page 582)
    Thus we have completed one mode of arriving at the result, and we shall now pass on to the other. If we proceed as in the latter part of Art. 1007 we shall find that the probability that the error in the value of x, when it is determined by (5), lies between t and t + dt is ^(SviOi)2 e 4*2 dt. (11). For put c — t] in equation (4) of Art. 1002. Then the proba- bility that will lie between 0 and 2r\ =— r 2k^tt J _r (1-V+V)° L_ f 2 k\J 7r J ( 2tj _ (l~v)2 4k2 dv. 4x2 dv 7] ZK>\J 7T J o Thus the probability that £7^ will lie between r and r + dr is 1 (*-*)» e 4x2 dr, 2.K\J 7T and therefore the probability that Xy^i will lie between l + r and l + t + dr is 1 e 4x2 c?r. This is therefore the probability that the error in the value of x when determined by (5) will lie between r , t + dr ^ — and ■ ^ • ^7 iai ^7iai And therefore the probability that the error in the value of x when determined by (5) will lie between t and t + dt is given by (11). The mean value of the positive error to be apprehended in the value of x will be obtained by multiplying the expression in (11) by t and integrating between the limits 0 and oo for t. Thus, since Vy.a. — i} we obtain for the result; and therefore if we pro- V 7T ceed to make this mean error as small as possible we obtain the same values as before for the factors yL, y2, 73, ... It will be interesting to develop the value of k. Multiply equation (7) by yu and sum for all values of i; thus by (6) we obtain K* = X.