# 28.1 Discrete random variables

## 28.1.1 The linearity of expectation

###### Example 28.1.1

Let $X$ be the sum of three fair dice, and let $A$ be the outcome \say$X$ is even. Then $\operatorname{E}[X|A]=\operatorname{E}[A]$.

This is true by the linearity of expectation, let $X_{1},X_{2},X_{3}$ be the respective dice rolls, then we know that

$\displaystyle\operatorname{E}[X|A]$ | $\displaystyle=\operatorname{E}[X_{1}+X_{2}+X_{3}|A]$ | (28.1) | ||

$\displaystyle=\operatorname{E}[X_{1}|A]+\operatorname{E}[X_{2}|A]+% \operatorname{E}[X_{3}|A]$ | (28.2) |

We have three cases here, but they are all symmetrical, so we can just consider

$\displaystyle\operatorname{E}[X_{3}|A]$ | $\displaystyle=\sum_{1\leq i\leq 6}i\times\Pr(X_{3}=i|X\text{ is even})$ | (28.3) |

Then of course the question is what is $\Pr(X_{3}=i|X\text{ is even})$. Here we can apply the definition of conditional probability

$\displaystyle\Pr(X_{3}=i|X\text{ is even})$ | $\displaystyle=\Pr\left(\frac{X_{3}=i\cap X\text{ is even}}{X\text{is even}}\right)$ | (28.4) |

and the easy thing to compute here is the probability that $X$ is even which is $\frac{1}{2}$. Then the slightly harder thing to compute is the numerator; clearly $X_{3}=i$ with probability

## 28.1.2 Variance of a discrete random variable

The "variance" of a discrete random variable^{1}^{1}
1
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.

^{2}

^{2}2 If it’s not clear why $X-\operatorname{E}[X]$ gives the signed distance between $X$ and $\operatorname{E}[X]$, take a look at the ”vectors” chapter.

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:

$\displaystyle\operatorname{Var}[X]$ | $\displaystyle=\operatorname{E}\left[\left(X-\operatorname{E}[X]\right)^{2}\right]$ | (28.6) | |||

$\displaystyle=\operatorname{E}\left[X^{2}-2X\operatorname{E}[X]+\operatorname{% E}[X]^{2}\right]$ | (step 1) | (28.7) | |||

$\displaystyle=\operatorname{E}[X^{2}]-2\operatorname{E}[X]\operatorname{E}[X]+% \operatorname{E}[X^{2}]$ | (step 2) | (28.8) | |||

$\displaystyle=\operatorname{E}\left[X^{2}\right]-\left(\operatorname{E}[X]% \right)^{2}$ | (28.9) |

When we went from step 1 to step 2, we took advantage of the fact that
$\operatorname{E}[X]$ is constant; in effect, we grouped our expression as$\operatorname{E}\left[\left(2\operatorname{E}[X]\right)X\right]$,^{3}^{3}
3
Bear in mind that $\operatorname{E}[X]$ is a constant and then used the
linearity of expectation
^{4}^{4}
4
If this means nothing to you, please be aware
that I have yet to write this section.
to rewrite it as
$\left(2\operatorname{E}[X]\right)\operatorname{E}[X]$.