Let f(x) be a continuous function defined for 0 ‰¤ x ‰¤ 1. Consider the functions

(called Bernstein polynomials) and prove that

Let X1, X2, . . . be independent Bernoulli random variables with mean x. Show that
Bn(x) = E[f(X1 + · · · + Xn / n)]
and then use Theoretical Exercise 4.
Since it can be shown that the convergence of Bn(x) to f (x) is uniform in x, the preceding reasoning provides a probabilistic proof of the famous Weierstrass theorem of analysis, which states that any continuous function on a closed interval can be approximated arbitrarily closely by a polynomial.

Transcribed Image Text:

lim Bn(x) f(x)