3.3 The binomial distribution  3. Probability Distribution Functions  3.1 Introductory comment

3.2 Gaussian or normal distribution

This distribution occurs frequently and has great generality. For large numbers of events, it is the limiting form for many other distribution functions, and by virtue of the central limit theorem it is the appropriate form for the sum of many variables even if those variables individually follow other distributions. It is

$$\Phi_G(x;\mu,\sigma) = {{1}\over{\sqrt{2\pi}\sigma}}\exp\{-{{(x-\mu)^2}\over{2\sigma^2}}\} .$$ (3.2)

 

The Gaussian distribution provides a realistic approximation to the distribution of deviations in many experimental situations, especially for the "central" portion of the deviations. The distribution function is plotted in Fig. 3.1  .The width of the distribution is characterized by the standard deviation $\sigma$, or sometimes by the full-width-at-half-maximum, $\Gamma=2.354\sigma$. See Fig. 3.2 for examples with various widths.

 

Figure 3.1: Frequency distribution for the Gaussian distribution function $\Phi_G$ as a function of the normalized deviation $x/\sigma$, for the case with zero mean value.

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Figure 3.2:  Gaussian probability distribution, as a function of the unnormalized deviation, for cases where $\sigma$ assumes the values 0.5, 1.0, 2.0, and 5.0.

 

   
 3.3 The binomial distribution  3. Probability Distribution Functions  3.1 Introductory comment 

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