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Welcome to your variance interview! In a quantitative finance interview, one of the most important things is to explain your thoughts. A good thought process is worth more than a correct answer with no ability to articulate your reasoning. This quiz is no different, as we’ll be breaking down a challenging problem into its logical steps.

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?

It makes sense to set indicator variables for the potential “successful” events. How many of these indicator variables should you have?

If we let \(X_k\) be an indicator variable on the event that flips \(k\) and \(k+1\) are both heads, what is \(\text{var}(X_k)?\)

In the previous question, we found that \(\text{var}(X_k) = \frac{3}{16}.\) Since the total payout is \(\sum_{k=1}^{9}X_k,\) is it true that the variance of the payout is \(9 \cdot \frac{3}{16} = \frac{27}{16}?\)

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, \[E\left(\sum_{k=1}^9 X_k\right)?\]

We’ve got the variance down to \[E\left(\sum_{k=1}^{9}X_k^2 + \sum_{k \ne j} X_kX_j \right) - 2.25^2.\] What is \[E\left(\sum_{k=1}^{9}X_k^2\right)?\]

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.

What is the variance of the payout?


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