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Variance

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?

Consider the data set $$\{x_1,x_2,\cdots, x_n\}$$ with variance 6. What is the variance of the data set $$\{2x_1, 2x_2,\cdots, 2x_n\}$$?

Consider the data set $$\{x_1,x_2,\cdots, x_n\}$$ with variance 6. What is the variance of the data set $$\{x_1+2, x_2+2,\cdots, x_n+2\}$$?

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 results from rolling a single fair six-sided die and multiplying the result by 2, and let $$B$$ be the variance of the number that results from rolling two fair six-sided dice and summing the results. Find the ordered pair $$(A,B)$$.

Suppose you flip a fair coin, and if it comes up heads, roll a fair six-sided die, and if it comes up tails, roll a fair four-sided dice. What is the variance of the number you obtain from this process?

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