B Probability Distribution Descriptions for Strategic Modeling Simulations

This appendix explains probability and probability distributions to help you select the most appropriate probability distribution for your Strategic Modeling simulation.

For each uncertain input in a simulation, you define the possible values with a probability distribution. The type of distribution you select depends on the conditions surrounding the input. A simulation calculates numerous scenarios of a model by repeatedly picking values from the probability distribution for the uncertain inputs and using those values to calculate the model.

To select the correct probability distribution:

1. Evaluate the input in question. List everything you know about the conditions surrounding this input. For example, you can gather valuable information about the uncertain input from historical data.
2. Review the descriptions of the probability distributions. This appendix describes each distribution in detail, outlining the conditions underlying the distribution. As you review the descriptions, look for a distribution that features the conditions you have listed for this input.
3. Select the distribution that characterizes this input, where the conditions of the distribution match those of the input.

Normal

The Normal distribution describes many phenomena such as returns on equity or assets, inflation rates, or currency fluctuations.

Decision-makers can use the normal distribution to describe uncertain inputs such as the inflation rate or periodic returns on assets.

Parameters

• Mean
• Standard Deviation

Note:

Of the values of a normal distribution, approximately 68% are within 1 standard deviation on either side of the mean. The standard deviation is the square root of the average squared distance of values from the mean.

Conditions

Use the normal distribution under these conditions:

• Mean value is most likely.
• It is symmetrical about the mean.
• It is more likely to be close to the mean than far away.

Triangular

The Triangular distribution describes situations where you know the minimum, maximum, and most likely values. In the simulation, the minimum and maximum values will never actually occur because their probability is zero.

It is useful with limited data in situations such as sales estimates, inventory numbers, and marketing costs. For example, you could describe the number of cars sold per week when past sales show the minimum, maximum, and usual number of cars sold.

Parameters

• Minimum
• Likeliest
• Maximum

Conditions

Use the triangular distribution under these conditions:

• Minimum and maximum are fixed.
• It has a most likely value in this range, which forms a triangle with the minimum and maximum.

Uniform

The Uniform distribution describes situations where you know the minimum and maximum values and all values are equally likely to occur.

Parameters

• Minimum
• Maximum

Conditions

Use the uniform distribution under these conditions:

• Minimum is fixed.
• Maximum is fixed.
• All values in range are equally likely to occur.

Lognormal

The Lognormal distribution describes many situations where values are positively skewed (where most of the values occur near the minimum value) such as asset and security prices. Such quantities exhibit this trend because values cannot fall below zero but can increase without limit.

Parameters

• Location
• Mean
• Standard Deviation

Note:

If you have historical data available with which to define a lognormal distribution, it is important to calculate the mean and standard deviation of the logarithms of the data and then enter these log parameters. Calculating the mean and standard deviation directly on the raw data does not give you the correct lognormal distribution.

Conditions

Use the lognormal distribution under these conditions:

• Upper and lower limits are unlimited, but the uncertain input cannot fall below the value of the location parameter.
• Distribution is positively skewed, with most values near the lower limit.
• Natural logarithm of the distribution is a normal distribution.

BetaPERT

The BetaPERT distribution describes situations commonly used in project risk analysis for assigning probabilities to task durations and costs. It is also sometimes used as a smoother alternative to the triangular distribution.

It describes a situation where you know the minimum, maximum, and most likely values to occur. It is useful with limited data. For example, you could describe the number of cars sold per week when past sales show the minimum, maximum, and usual number of cars sold.

Parameters

• Minimum
• Likeliest
• Maximum

Conditions

Use the betaPERT distribution under these conditions:

• Minimum and maximum are fixed.
• It has a most likely value in this range, which forms a triangle with the minimum and maximum; betaPERT forms a smoothed curve on the underlying triangle.

Yes-No

The Yes-No distribution describes situations that can have only one of two values: for example, yes or no, success or failure, or true or false.

Parameters—Probability of Yes

Conditions

Use the yes no distribution under these conditions:

• For each trial, only 2 outcomes are possible, such as success or failure; the random input can have only one of two values, for example, 0 and 1.
• The mean is p, or probability (0 < p < 1).
• Trials are independent. Probability is the same from trial to trial.