/*   9-27-2023   mathematica expectation R(f')   */

     /* BCV  */

fx = f0 + x f1 + x^2 f2 /2 + x^3 f3/6 + x^4 f4 /24

p0 = Integrate[ fx,{x,-h,0}]

p1 = Integrate[ fx,{x,0,h}]

ans = Expand[ (p1-p0)^2 / (h^3) ]



       /*  UCV    */


pA = Normal [ Series[ p1, {h,0,4} ] ]

ans$ucv = n*pA*(1-pA) + (n*pA)^2

ucv = 2/((n-1)*h) - (n+1)*ans$ucv / (n^2 * (n-1) * h )


ucv$alt = 2/(n*h) - ans$ucv / (n^2 * h )

Expand[ ucv$alt ]