Tuesday, May 9, 2017

Binomial Distribution Fit Curvature

If one takes the natural log of the probabilities for the binomial distribution and the fit in the last blog one gets the curves below. The 2nd differences which are a measure of the curvature of the curves are also given. The 2nd differences for the fit are constant as expected.

The relatively large differences at the ends are less critical since they correspond to relatively small values for the probabilities.

Supplemental (May 9): The 2nd differences for the normal distribution function are also uniform and equal to -0.04 or 1/λ exactly.

Supplemental (May 10): Technically, curvature depends on changes in the tangent of a curve with path length but I think it's fair to say that deviation from a straight line is a form of curvature even if it is not constant. For a parabolic arc the rate of change of the slope with a change in the "horizontal" distance is constant. I got Excel to find a center for the circular arc of the lower curve of the second plot above and radius of curvature turned out to be a little over 25,000. We don't have to worry about units here since both axes are just real numbers.