Understanding PERT Formula Proof for Project Management

[Tawoos]

a formula used in project management for determining the expected time of an activity states that:

Expected time = (Pessimistic value + 4*most likely value + Optimistic value)/6​

I am searching for a proof for this law. I've searched the internet, and found out it's related to the Beta Distribution function, but little details about how the shape parameters of the beta distribution were chosen and why?

Can someone explain to me the outline of the proof at least?

Thanks in advance.​

 

[HallsofIvy]

That is certainly not a "law" but rather an approximation or estimate. Mathematically, it is simply a "weighted average". I presume that the choice of parameters is based on experience in the field, not mathematics.

 

[mXSCNT]

It looks like it might also have something to do with Simpson's rule. Let f(x) be your estimate of the xth percentile for the time (so e.g. f(50%) is your estimate of the median time required). Suppose your "low estimate" is f(25%), your "medium estimate" is f(50%), and your "high estimate" is f(75%). Simpson's rule tells you that the average of f over the interval (25%, 75%) can be approximated by (f(25%) + 4f(50%) + f(75%))/6. I don't know what good that average does you; perhaps it is somehow related to the expected time.