Figure 8.6 Tusler's Risk Classification Scheme

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Continuous Probability Distributions Continuous probability distributions are useful for developing risk analysis models when an event has an infinite number of possible values within a stated range. Although in theory there are an infinite number of probability distributions, we will discuss three of the more common continuous probability distributions used in modeling risk. These include the Normal Distribution, the PERT distribution, and the triangular distribution. A quick overview shows how these distributions may he used to develop models for simulation or sensitivity analysis.

One of the most common continuous probability distributions is the normal distribution, or Bell Curve. Figure 8.8 provides an example of a normal distribution. The normal distribution has the following properties:

■ The distribution's shape is determined by its mean (ji) and standard deviation (cr). In Figure 8.8, this particular distribution has a mean of 0 and a standard deviation of I. Other combinations of means and standard deviations will result in normal distributions with shapes that are either flatter or taller.

■ Probability is associated with area under the curve. Therefore, the total area under the curve and the baseline that extends from negative infinity (— to positive infinity (+ is 100 percent. Subsequently, to find the probability of an event occurring between any two points on the baseline, just find the area between those two points under the curve. This is done by standardizing a given value for X using the formula: z = (X — |i) cr to obtain a z score. A table with the various z scores is then used to compute the probability for the area between any two z scores.

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0.45 ■ 0.40 0.35 0.30 0.25 ■ 0.20 0.15 0.10 0.05 0.00

-1.6448 Figure 16448

■ Since the normal distribution is symmetrical around the mean, an outcome that falls between - <* and the mean, ft, would have the same probability of falling between the mean, p. and + » (i.e., 50 percent).

■ Since the distribution is symmetrical, the following probability rules of thumb apply

About 68 percent of all the values will fall between ±la of the mean About 95 percent of all the values will fall between ±2o of the mean About 99 percent of all the values will fall between ±3a of the mean

Therefore, if we know or assume that the probability of a risk event follows a normal distribution, we can predict an outcome with some confidence. For example, let's say that a particular project task has a mean duration of ten days. Moreover, over time we have been able to determine that this particular task has a standard deviation of two days. The mean tells us that if we were to complete this particular task over and over again, we would expect to complete this task, on average, in ten days. If we always completed the task in ten days, there would be no variability and the standard deviation would be zero. If, however, the task sometimes took anywhere between six and fifteen days to complete, we would have some variability, and the standard deviation would be a value greater than zero. The more variability we have, the larger is the computed standard deviation.

Using the rules of thumb described above, we could estimate, for example, that we would be about 95 percent certain that the project's task would be complete within six to fourteen days (p ± 2 cr = 10 ± 2 * 2). In addition, we could also say that we would be about 99 percent confident that the task would be completed between four and sixteen days (p ± 3<t = 10 ± 3 x 2).

PERT Distribution Using the PERT distribution, one can find a probability by calculating the area under the curve. However, the PERT distribution uses a three-point estimate where:

■ ci denotes an optimistic estimate

■ b denotes a most likely estimate

■ c denotes a pessimistic estimate

Therefore, the mean for the PERT distribution is computed using a weighted average as follow:

PERT Mean = (a + 4m + &) + 6 And the standard deviation is computed:

PERT Standard Deviation = (b-a)+ 6 Figure 8.9 provides an example of a PERT distribution where a = 2, m = 4, and b = 8.

Triangular Distribution Lastly, the triangular distribution, or TRIANG, also uses a three-point estimate similar to the PERT distribution where:

■ a denotes an optimistic estimate

■ b denotes a most likely estimate

■ c denotes a pessimistic estimate

However, the weighting for the mean and standard deviation are different.

TRIANG Mean = (at + m 4- b) + 3 TRAING Standard Deviation = [((ft - a)2 + (m-a)(m - b)) - 18]"2

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