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dx. (Normal or Gaussian (6.3) distribution)

This expression is characterized by two parameters, m (the mean which is the true value of the quantity being measured) and aw (a width parameter for now), whereas the Poisson distribution is described with the single parameter m. We introduce (3) solely as a mathematical function but its physical utility will soon become clear. We will find below that aw is also the standard deviation a of the distribution. The subscript "w" is used here to distinguish temporarily the two quantities.

The normal distribution is the well known bell curve of probability; two examples are shown in Fig. 8 (smooth curves). It is symmetric about m and can extend to negative values of x. The coefficient (2^)-1/2 is chosen so that this distribution is also normalized; that is, the integral of all probabilities is unity, xx x d Px = 1 (6.4)

where the integral limits extend to all possible values of x.

The quantity aw in (3) describes a characteristic width of the distribution; at x = m ± aw, the function has fallen to e-0 5 = 0.601 of its maximum value. It falls

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