Calculating probabilities of occurrence

A statistical distribution is a function that describes the behavior of a random variable, that is, a quantity that can assume any permissible value for the system to which it refers, but for which the chance of occurrence is governed by some probabihty distribution. If we could discover or estimate the nature of this distribution, we could calculate the probability of occurrence of any value of interest. We would, in fact, possess a sort of statistical crystal ball we could use to make predictions. Soon we will see how to do this using the normal distribution. 

The normal distribution is a continuous distribution, that is, a distribution in which the variable can assume any value within a predefined interval. For a normally distributed variable, this interval is (- 00, + 00), which means that the variable can — in principle — assume any real value. 

A continuous distribution of the variable x is defined by its probability density function fix), a mathematical expression containing a certain number of parameters. The normal distribution is fully defined by two parameters, its mean and its variance (Eq. (2.5)). 

To indicate that a random variable x is normally distributed with mean /i and variance we use the notation x where the notation  

Note that the curve is perfectly symmetrical about its central point, the mean fi (here equal to zero). The density reaches its maximum value at the mean, and rapidly decreases on both sides as we move away from it. At three standard deviations from the mean, the probability density becomes almost zero. These features are similar to those observed in the histogram of the masses of the 140 beans shown in Fig. 2.2. 

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When the probabilities of occurrence of primary/basic fault events are known, the occurrence probability of the top event can be calculated. This can only be achieved by first calculating probabilities of occurrence of the output fault events of all the intermediate and lower logic gates (e.g., AND and OR gates). 

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