Using Estimated Expected Values to Fit a Trapazoidal Distribution

 From the definition of the trapazoidal distribution and its integral one can obtain formulas for the expected values ⟨x⟩ and ⟨x²⟩.

Then given a set of random numbers from a trapazoidal generator one can analyze the set by counting the number of values that fall within chosen intervals and then estimate the distribution for the intervals and the expected values ⟨x⟩ and ⟨x²⟩.

We now have two equations which can be solved for a and b which can be used to fit the observations and compare the results with the original values of a and b. In the example below the original values were a=0.5 and b=1.0 and the fit values were a=0.538 and b=0.993.

This was a lot easier to do since the equation for ⟨x⟩ can be transformed into a quadratic function of ρ=a/b whose solution can be found if a set of values are assumed for b.  The set of values for ρ can then be used to find a zero of the expression for ⟨x²⟩.