A. The triangular distribution

A triangular distribution is named by the graph of it's density function, that looks like fig:A-triangular-density, where points , and are respectively the mode and of the distribution. Function is given fig:A-triangular-cdf, while mean and variance can be obtained at first sight according to the mechanical behavior of a triangular plate. Namely :

Fig. 5: Triangular density.

Fig. 6: Triangular cdf.

The most important reason to use the triangular distribution is its simplicity. Being easy to use, this distribution is a good candidate when you want to check if your conclusions remain valid when another distribution is used that fits with what knowledge you have extracted from your data.

Compared with a Gaussian model that allows only to ensure , the triangular distribution allows to reach . Moreover, the demand has no reason to be symmetrical around it's mean. This lack of symmetry is usually measured by the , where is the third centered moment. This moment has a nice expression over  :

But we can obtain a more compact expression by using (the mean), (the width) and (the barycentric position of in , verifying ) to characterize the distribution. We obtain

and therefore the skewness depends only on . Conversely, the skewness gives you (the shape), then gives you (the size) and finaly fixes the position of the triangle along the axis. By this method, you can deal with skewness up to .