1 Studies in the History of Statistics and Probability ... - Sheynin, Oscar

(1/B n )(m − A n )instead of the value of the sum m = (ξ 1 + ...+ ξ n ) will be necessary.And now the essence of Laplace’s discovery can be expressed by asingle phrase: the figure should be shifted byA n = E(ξ 1 + ... + ξ n ) = naand the coefficient of the change of the scale should be equal toBn= var(ξ + ... + ξ ) = σ n.1nThe random variable* 1sn= [(ξ1+ ... + ξn) − E(ξ1+ ... + ξn)]var(ξ + ... + ξ )1n(4.6)is called the normed sum. Obviously,* *Esn= 0, var sn= 1and the numbers (4.5b) are the possible values of that normed sum. Letus attempt to show its probabilities as rectangles with bases1xn( m + 1) − xn( m) = ,σ ntheir midpoints coinciding with points x n (m) and areas equal toprobabilitiesP s x m P m*{n=n( )} = {ξ1+ ... + ξn= }.The heights of these rectangles should be*σ nP{ sn = xn ( m)} = σ nP{ξ 1+ ... + ξn= m}.Thus, because of (4.5a) the upper bases of these rectangles will bealmost exactly situated along a curve described by equation21 xy = y( x) = exp[ − ](4.7)2π 2independent of anything and calculated once and for all.A result absolutely not foreseen and almost miraculous! Disorder inprobabilities p m somehow gives birth to a unique curve (4.7) whichsimply occurs by summing random variables and transforming thescale of the figure. That is Laplace’s remarkable discovery without33