Variance measures the "spread" of a random variable. More specifically, \[\text{var}(X) = E\left(\big(X-E(X)\big)^2\right).\]

Through expansion and simplification, this can also be written as \[\text{var}(X) = E\big(X^2\big) - E(X)^2.\]

Standard deviation is just the square root of variance. Both of these ideas are used widely in quantitative finance in order to model the fluctuations of assets and to quantify risk.

Which of the following always has the same units as the data it measures?

What is the variance of the number that results from a single roll of a fair six-sided dice?

There are two ways to imagine combining the variances of the single dice roll: rolling a single die, and adding that number to itself, or rolling two dice and adding the results. Do you think these have the same variance?

This is analogous to the difference between buying two shares of one stock versus buying one share of two similar (but independent) stocks.

Let \(A\) be the variance of the number that comes from finding 2 times the value obtained rolling a single fair six-sided die.

Let \(B\) be the variance of the number that comes from rolling two fair six-sided dice and summing the results.

Find the ordered pair \((A,B)\).

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