ABSTRACT

Bernstein polynomials are known to approximate any continuous function on a 
closed and bounded interval.  The approximating polynomial converges uniformly to 
the function as the degree, n, of the polynomial goes to infinity.  Furthermore, 
any derivative of the function is approximated by the corresponding derivative of 
the approximating polynomial, which also converges uniformly as n goes to 
infinity.

In this talk, it is shown how to make use of Bernstein polynomials to approximate 
the moments of a continuous function, g(X), of a random variable, X. It is 
demonstrated that this approach provides a better approximation than the 
traditional delta method, which uses a first-order Taylor's series approximation 
of g(x) and requires that g(x) be differentiable around the mean of X.  By 
contrast, Bernstein polynomials only require the function g(x) to be continuous, 
but not necessarily differentiable.   Two numerical examples are presented to 
demonstrate the application of Bernstein polynomials in approximating moments.