Bernstein Polynomials and Expected Value The first equation in this paper
http://www.emis.de/journals/BAMV/conten/vol10/jopalagzyl.pdf
is:
$$\displaystyle B_nf(x)=\sum_{i=0}^{n}\binom{n}{i}x^i(1-x)^{n-i}f\left(\frac{i}{n}\right)=\mathbb E f\left(\frac{S_{n,x}}{n}\right)$$
where $f$ is a Lipschitz continuous real function defined on $[0,1]$, and $S_{n,x}$ is a binomial random variable with parameters $n$ and $x$.
How is this equation proven?

$$\mathbb E(g)=\sum_{i}\mathbb P\left(S_{n,k}\right)g(i)$$ 
where:
$$g(S_{n,x})=f\left(\frac{S_{n,x}}{n}\right)$$
Therefore we have:
$$\mathbb E\left(f\left(\frac{S_{n,x}}{n}\right)\right)=\sum_{i=0}^{n}\mathbb P(S_{n,x}=i)f\left(\frac{i}{n}\right)$$
But $$\mathbb P(S_{n,x}=i)=\binom{n}{i}x^i(1-x)^{n-i}$$
Therefore
$$\sum_{i=0}^{n}\binom{n}{i}x^i(1-x)^{n-i}f\left(\frac{i}{n}\right)=\mathbb E\left(f\left(\frac{S_{n,x}}{n}\right)\right)$$ which was the original statement.
