The Normal Gaussian Distribution

The normal distribution, sometimes also called the Gaussian distribution, is a continuous distribution that is very commonly used. Its shape is as shown in Figure 11.1(a), the familiar bell-shaped curve.

One common formula for the density function for the normal distribution, shown in Figure 11.1, is

The normal distribution (in this case) is symmetric about x = 0 and has the standard deviation ct as a parameter.

If we wish to calculate probabilities, however, it is necessary to integrate a continuous density function. Formally, the cumulative distribution function (CDF) is found, for the continuous case, as

F(x) calculates the probability that random variable x is equal to or less than some particular value. For example, in Figure 11.1, we would calculate the probability that the random variable is less than or equal to zero as 0.5 by integrating from minus infinity to zero. The CDF for the normal distribution with zero mean value is also plotted in Figure 11.1(b), and its ordinate values correspond directly to probabilities.

The normal distribution is not readily integrable, so that we resort to table lookups in order to calculate other than the simplest probability cases. Further, these tables have been developed for the ''standard normal,'' which has a mean value of zero and a standard deviation of unity.

(a) The density function

Figure 11.1. The normal (Gaussian) distributing.

(a) The density function

(b) The cumulative distribution function (CDF)

Figure 11.1. The normal (Gaussian) distributing.

Example. For the normal distribution shown in Figure 11.1, the probabilities that the random variable is equal to or less than one and two sigma (ct) to the right of the mean are:

For one sigma, the value is 0.5 + 0.3413 = 0.8413 For two sigma, the value is 0.5 + 0.4772 = 0.9772

The values 0.3413 and 0.4772 were obtained by referring to a standard normal table for the argument equal to one and two, respectively. The probabilities that the variable lies within plus-and-minus one and two sigma from the mean are simply 0.6827 and 0.9545, respectively. Thus, about 95% of the distribution lies within plus-and-minus two sigma from the mean.

Example. We assume that the critical path of a simple PERT network consists of five independent sequential activities, with the following estimates of optimistic, most likely, and pessimistic times in weeks:

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