problem 53

4.3.13 problem 53

Let $I_{j}$ be the indicator variable for tosses $j$ and $j + 1$ landing heads for $1 \leq j \leq 3$ . Then, the expected number of such pairs is $E (X) = \sum_{j = 1}^{3} P (I_{j} = 1) = 3 p^{2}$ . $Var (X) = E (X^{2}) - 9 p^{4}$ .

$E (X^{2}) = E ({(\sum_{j = 1}^{3} I_{j})}_{j}^{2}) = E ({(I_{1} + I_{2})}_{1}^{2} + 2 (I_{1} + I_{2}) I_{3} + I_{3}^{2}) = E (I_{1}^{2} + 2 I_{1} I_{2} + I_{2}^{2} + 2 I_{1} I 3 + 2 I_{2} I 3 + I_{3}^{2})$ .

Note that $I_{j}^{2} = I_{j}$ .

Note that $E (I_{1} I_{2}) = E (I_{2} I_{3}) = p^{3}$ as these require 3 consecutive heads to equal 1, but $E (I_{1} I_{3}) = p^{4}$ as this requires 4 consecutive heads to equal 1. Thus, $E (X^{2}) = (p^{2} + 2 p^{3} + p^{2} + 2 p^{4} + 2 p^{3} + p^{2}) = 4 p^{3} + 3 p^{2} + 2 p^{4}$ .

Thus,

Var (X) = 4 p^{3} + 3 p^{2} - 7 p^{4}

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