The triangular distribution is a useful tool if a variable has to be estimated subjectively. The estimator has to indicate a Low, a Most Likely value (Mode) and a High value, the distribution contained within the Low to High range. In the formulas below "l" is the Low, "m" is the mode and "h" the High value. In other descriptions (Wikipedia) the characters "a", "c" and "b" are used resp. for l, m and h. The pdf is a triangle:
The mean is:
In this case it is interesting to know also the inverse form of the cumulative distribution function:
This formula is used in generating a random triangular deviate from a rectangular one between 0 and 1 in Monte Carlo analysis.
However, this simple solution may go entirely wrong if the distribution of the data is such that the mean is very low in comparison to the
High parameter. Then the mode estimate might end up on the left of the Low! In such case the triangular distribution can not reasonably be fitted to the data.
Truncated triangular distribution
In prospect appraisal, it is sometimes useful to describe an unrisked volume distribution as a triangular. When an economic minimum volume is given, the original triangular will become truncated from the left, at a cutoff-volume "c". Then the MSVc, i.e. the mean success volume after cutoff has to be calculated. Also the POSc, or the probability of success after cutoff is required. The latter is the total risk: geological POSg + the risk to find HC, but less than the cutoff. These parameter, and their product (Ec = POSc times MSVc) can be analytically calculated.