## Info

Figure 6.8. The Poisson (step curve) and normal distributions (smooth curves) for the mean value m = 100. The normal distribution is given for two values of the width parameter aw which is shown in the text to be equal to the standard deviation a. The Poisson distribution approximates well the normal distribution if the latter has a = *Jm. Note the slight asymmetry of the Poisson distribution relative to the normal distribution. The standard deviation and full width half maximum widths are shown for the higher normal peak; the two normal curves happen to cross at the FWHM point.

to e-1 = 0.37 at x — m = -J2 aw, i.e., whenx is removed from m by aw. These widths indicate the spread of the function about the mean. The full width at half maximum (FWHM) can be shown to be 2.36aw (see Fig. 8).

One can show by integration of (3) that 68% of the area under the distribution falls within m ± aw. That is, the probability of a measurement falling within 1aw of the mean is 0.68, or equivalently the probability of falling outside these limits is 0.32 (Table 2). The probability that the result of a measurement will fall outside m ± 2aw is 0.046 or 4.6%, or that it will fall outside m ± 3aw is 0.0027 or 0.27%. It is thus rather unlikely that a single measurement would yield a fluctuation of 3aw from the mean. It is extremely unlikely (one chance in two million) that a 5aw

Table 6.2. Normal distribution probabilities

Table 6.2. Normal distribution probabilities

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