28.1 Discrete random variables

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

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

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

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

The "variance" of a discrete random variable¹¹ 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.

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:

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$\operatorname{E}\left[\left(2\operatorname{E}[X]\right)X\right]$,³³ 3 Bear in mind that $\operatorname{E}[X]$ is a constant and then used the linearity of expectation ⁴⁴ 4 If this means nothing to you, please be aware that I have yet to write this section. to rewrite it as $\left(2\operatorname{E}[X]\right)\operatorname{E}[X]$.