problem 40

4.3.3 problem 40

Let $I_{j}$ be the indicator random variable for HTH pattern starting on the $j$ -th toss. Since the tosses are independent, $P (I_{j} = 1) = \frac{1}{8}$ for $1 \leq j \leq n - 2$ .

Let $X = \sum_{j = 1}^{n - 2} I_{j}$ be the number of HTH patterns in $n$ independent coin tosses. Then,

E (X) = E (n - 2 \sum j = 1 I_{j}) = n - 2 \sum j = 1 E (I_{j}) = n - 2 \sum j = 1 \frac{1}{8} = \frac{n - 2}{8}

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