PERT MoM Fit Details

Property 1: If a data set X follows a PERT distribution, we can estimate the distribution parameters a, b, and c by setting the formulas for the distribution mean, standard deviation, and skewness to the corresponding sample values m, s, and w. In particular

Estimating c

Proof: First, we observe that the skewness of the data in X (see PERT Distribution).

Skewness for Beta distribution

where

Alpha and beta values

Note that

Thus

It now follows that

As we observed in Method of Moments: PERT Distribution

Hence

Also

Thus, the numerator of the expression for w2 is equal to

We also know from Method of Moments: PERT Distribution that a = c – 6s and so a2 = c2 -12cs + 36s. Thus, the numerator of the formula for w2 is equivalent to

We also observe that the denominator of the formula for w2 is equivalent to

Putting the pieces together, we see that

We now solve for c.

This is equivalent to the quadratic equation

where

In fact, if we also define

then, the original equation can be expressed as

Using the quadratic formula, we get

which completes the proof.

References

Rao, K. S., Viswam, N., Anjaneyulu, G. (2021) Estimation of parameters of Pert distribution by using method of moments
https://www.ijraset.com/fileserve?FID=38239

Wikipedia (2023) PERT distribution
https://en.wikipedia.org/wiki/PERT_distribution

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