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Double-triangular Distribution

Double-triangular Distribution is the combination of two triangles, each with an area of 0.5. The mode is also the median.

The mean is:

The variance is:

Note: the variance is the same as for the triangular distribution.

The probability density is:

And the cumulative distribution function:

 

For Monte Carlo simulation random values from the DT can be generated using random numbers between 0 and 1 (here denoted as “p”) and the following formulas:

The double-triangular is quite unnatural. It is highly unlikely to be a proper representation of uncertainty. Moreover, estimating the median is, in my view, more difficult than the estimating the most likely value (mode).

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three-point estimation

The three-point estimation technique is used in management and information systems applications for the construction of an approximate probability distribution representing the outcome of future events, based on very limited information. While the distribution used for the approximation might be a normal distribution, this is not always so and, for example a triangular distribution might be used, depending on the application.,[1]

In three-point estimation, three figures are produced initially for every distribution that is required, based on prior experience or best-guesses:

  • a = the best-case estimate
  • m = the most likely estimate
  • b = the worst-case estimate.

These are then combined to yield either a full probability distribution, for later combination with distributions obtained similarly for other variables, or summary descriptors of the distribution, such as the mean, standard deviation or percentage points of the distribution. The accuracy attributed to the results derived can be no better than the accuracy inherent in the 3 initial points, and there are clear dangers in using an assumed form for an underlying distribution that itself has little basis.

Based on the assumption (possibly unwarranted) that a double–triangular distribution governs the data, several estimates are possible. These values are used to calculate an E value for the estimate and a standard deviation (SD) as L-estimators, where:

E = (a + 4m + b) / 6
SD = (b − a) / 6

E is a weighted average which takes into account both the most optimistic and most pessimistic estimates provided. SD measures the variability or uncertainty in the estimate. In Project Evaluation and Review Techniques (PERT) the three values are used to fit a Beta distribution for Monte Carlo simulations.

The triangular distribution is also commonly used. It differs from the double-triangular by its simple triangular shape and the mode does not have to coincide with the median. The mean (expectation) is then:

E = (a + m + b) / 3.

In some applications,[1] the triangular distribution is used directly as an estimated probability distribution, rather than for the derivation of estimated statistics.