Discussion week8
Week 8, Sums of RV / Law of Large #’s
Fundamentals of Computational Mathematics
CUNY MSDS DATA 605, Fall 2018
Rose Koh
10/17/2018
Chapter 7.8
Section 8.1.17
For x ∈ [0, 1], let’s toss a biased coin that comes up heads with probability \(x\).
\(E (\frac{f(S_n)}{n})\) -> \(f(x)\)
However,
\(E(\frac{f(S_n)}{n})\) = \(\sum_{k=0}^{n} f(\frac{k}{n}) \binom{n}{k} x^k (1-x)^{n-k}\)
The left side tends to f(x) and the right side is polynomial.
Thus,
\(\sum_{k=0}^{n} f(\frac{k}{n}) \binom{n}{k} x^k (1-x)^{n-k}\) -> \(f(x)\)
This shows that we can obtain any continuous function \(f(x)\) on [0,1] as a limit of polynomial functions.