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Here’s the problem we’ll be tackling:

You are playing a game in which you will be paid $1 for each pair of consecutive heads you flip out of 10 coins. For example, if you flip THHHTTHHTT, you will receive $3. What is the variance of your payout?

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Right, we can’t add the variances of the \(X_k,\) since at least some of them are dependent. Using the identity \(\text{var}(X) = E(X^2) - \left(E(X)\right)^2,\) we can write the variance of the payout as \[E\left(\sum_{k=1}^{9}X_k^2 + \sum_{k \ne j} X_kX_j \right) - \left(E\left(\sum_{k=1}^9 X_k\right)\right)^2.\]

We can compute each part of this variance separately. For starters, what is the expected value of the game, \[E\left(\sum_{k=1}^9 X_k\right)?\]

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We’ve got the variance down to \[2.25 + E\left(\sum_{k \ne j} X_kX_j \right) - 2.25^2.\] What is \[E\left(\sum_{k \ne j} X_kX_j \right)?\]

**Hint:** There are a total of \(9 \cdot 8\) pairs \((X_k,X_j)\) in the sum. Some of these are pairs of independent random variables, while others are of dependent random variables.

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What is the variance of the payout?

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