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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2015

Paper 66

Every word in the <typical set> has probability at least $2^{-n(H(Z)+\delta)}$. Consequently
$$
1\geq\sum_{z^n\in T_\delta^n}P_Z^n(z^n)\geq|T_\delta^n|\,2^{-n(H(Z)+\delta)}.
$$
Rearranging proves the upper estimate in the <typical-set cardinality bounds>:
$$
\boxed{|T_\delta^n(P_Z)|\leq2^{n(H(Z)+\delta)}.}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 66 / 4 / ii / b / Solution

Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 66 / 4 / ii / b / Solution

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Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 66 / 4 / ii / b / Solution

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