A calculus of the absurd

27.1.2 Variance of a discrete random variable

The "variance" of a discrete random variable¹⁴³¹⁴³ Note: you’re not imagining things, I still need to add the section I have written defining these. is a measure of "spread" (how far apart values in a distribution are). It gives the expected value of the square of the distance of the observed values (in the outcome space) from the mean (expected value of the distribution). That’s a mouthful to say, so it can be easier to write this as a formula.

¹⁴⁴¹⁴⁴ If it’s not clear why \(X - \Expected [X]\) gives the signed distance between \(X\) and \(\Expected [X]\), take a look at the "vectors" chapter.

\begin{equation} \Var [X] = \Expected \left [\left (X - \Expected [X]\right )^2\right ] \end{equation}

There is an equivalent way in which the variance can be expressed which is a bit easier to use when trying to calculate the variance of a discrete random variable by hand:

\begin{align} \Var [X] & = \Expected \left [\left (X - \Expected [X]\right )^2\right ] \\ & = \Expected \left [X^2 - 2X\Expected [X] + \Expected [X]^2\right ] & \text {(step 1)} \\ & = \Expected [X^2] - 2\Expected [X]\Expected [X] + \Expected [X^2] & \text {(step 2)} \\ & = \Expected \left [X^2\right ] - \left (\Expected [X]\right )^2 \end{align}

When we went from step 1 to step 2, we took advantage of the fact that \(\Expected [X]\) is constant; in effect, we grouped our expression as\(\Expected \left [\left (2\Expected [X]\right )X\right ]\),¹⁴⁵¹⁴⁵ Bear in mind that \(\Expected [X]\) is a constant and then used the linearity of expectation ¹⁴⁶¹⁴⁶ If this means nothing to you, please be aware that I have yet to write this section. to rewrite it as \(\left (2\Expected [X]\right )\Expected [X]\).