|
HW6 (Due on 10/31)
(Typical sequences of coin-flips) Consider sequences of \(N\) coin-flips with the probability of head equal to \(p\). Denote \(n_H({\bf x})\) as the number of heads in a sequence \(\bf x\). Assume \(p < 0.5\). Show that a sequence \(\bf x\) is \(\epsilon\)-typical \(\left({\bf x} \in A_{\epsilon}^N(X)\right)\) if and only if
\( p- \frac{\epsilon}{\log\frac{1-p}{p}} \le \frac{n_H({\bf x})}{N} \le p+ \frac{\epsilon}{\log\frac{1-p}{p}}\)
Let \(p=0.2\), \(\epsilon=0.01\), and \(N=100\), compute \(Pr({\bf X}^N \in A_\epsilon^{N}(X))\)
Repeat (a) with \(N\) change to \(10,000\).
Repeat (a) and (b) with \(p\) change to \(0.45\).
|