Revision as of 02:27, 9 June 2024 by Bot (Created page with "<div class="d-none"><math> \newcommand{\NA}{{\rm NA}} \newcommand{\mat}[1]{{\bf#1}} \newcommand{\exref}[1]{\ref{##1}} \newcommand{\secstoprocess}{\all} \newcommand{\NA}{{\rm NA}} \newcommand{\mathds}{\mathbb}</math></div> Let us toss a biased coin that comes up heads with probability <math>p</math> and assume the validity of the Strong Law of Large Numbers as described in Exercise Exercise. Then, with probability 1, <math display="block"...")
BBy Bot
Jun 09'24
Exercise
[math]
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Let us toss a biased coin that comes up heads with probability [math]p[/math]
and assume the validity of the Strong Law of Large Numbers as described in Exercise Exercise. Then, with probability 1,
[[math]]
\frac {S_n}n \to p
[[/math]]
as [math]n \to \infty[/math]. If [math]f(x)[/math] is a continuous function on the unit interval, then we also have
[[math]]
f\left( \frac {S_n}n \right) \to f(p)\ .
[[/math]]
Finally, we could hope that
[[math]]
E\left(f\left( \frac {S_n}n \right)\right) \to E(f(p)) = f(p)\ .
[[/math]]
Show that, if all this is correct, as in fact it is, we would have proven that any continuous function on the unit interval is a limit of polynomial functions. This is a sketch of a probabilistic proof of an important theorem in mathematics called the Weierstrass approximation theorem.