Another interesting possibility is to consider a form of the function that satisfies the boundary conditions and has a satisfactory derivative.
We know that W has a positive derivative with a maximum at 0, and have every reason to believe that it will be well behaved. So casting about for a similar function, one familiar form of the derivative would be Aexp[-Bx^2]. Again, we see that I introduce a bias toward e.
Using the fundamental theorem of calculus
W(x)
= Integral[0:x] ( A exp[ -Bz^2 ] dz )
= A Integral[0:x] ( exp[ -Bz^2 ] dz )
= A/sqrt(B) Integral[0:x sqrt(B)] ( exp[-t^2] dt )
= A / sqrt(B) erf ( x sqrt(B) )
= A erf ( x sqrt(B) ) / sqrt (B) )W(1)
= 1/2
= A erf ( sqrt(B) ) / sqrt (B) )
I wasn't able to find a nice closed form, but you can choose a polynomial representation with a reasonable amount of accuracy (if you restrict yourself to the narrow band where baseball is actually played, for example, you can probably be satisfied with a simple quadratic).
July 29, 2004 9:19 PM
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