Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 224 / 2 / c

Table of contents
- Solution c

Solution

Conditionally on

X_{1}^{n} = x_{1}^{n}

, the uniform index

J

selects the symbol

a

with probability

P_{x_{1}^{n}} (a)

. The law of total probability therefore gives

\overline{P} (a) = P (X_{J} = a) = \sum_{x_{1}^{n} \in B} P_{n} (x_{1}^{n}) P_{x_{1}^{n}} (a) = \sum_{x_{1}^{n} \in B} \frac{Q ^{n} ( x _{1}^{n} )}{Q ^{n} ( B )} P_{x_{1}^{n}} (a) .

Thus

\overline{P}

is the conditional mean type of a string in

B

.

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