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

Isaac Todhunter

Date:: 1865

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

594/648 (page 574)

$observation the quantities of which 7^ is the type are all equal, and so are those of which 7q is the type. Thus (4) becomes u = IqA Tclqi + 2** (5); and the limits of error become 2 rli -7W)' if we suppose also that positive and negative errors are equally likely, we have 1c = 0, as in Art. 1004. Thus (5) be- comes (6). This agrees with Laplace’s result. Laplace also presents another view of the subject. Suppose that y\r (x) dx represents the chance that an error will lie between x and x + dx\ then I %yjr (x) dx may be called the mean value ' 0 of the positive error to be apprehended—la valeur moyenne de Terr ear d craindre en ‘plus. Laplace compares an error with a loss at play, and multiplies the amount of the error by the chance of its happening, in the same way as we multiply a gain or loss by the chance of its happening in order to obtain the advantage or disadvantage of a player. Laplace then examines how the mean value of the error to be apprehended may be made as small as possible. In equation (4) of Art. 1002 put c = rj; and suppose positive and negative errors equally likely, so that 7=0: then the proba- bility that will lie between 0 and 2y 1 pt - 2k \Itt J _v (v-y)3 iK‘ dv = l r2 2k \]TV J0 e iK’2 dv. Thus the probability that will lie between 0 and r is 2k *JttJ o$