28.1 Discrete random variables
28.1.1 The linearity of expectation
Example 28.1.1
Let
This is true by the linearity of expectation, let
(28.1) | ||||
(28.2) |
We have three cases here, but they are all symmetrical, so we can just consider
(28.3) |
Then of course the question is what is
(28.4) |
and the easy thing to compute here is the probability that
28.1.2 Variance of a discrete random variable
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:
(28.6) | |||||
(step 1) | (28.7) | ||||
(step 2) | (28.8) | ||||
(28.9) |
When we went from step 1 to step 2, we took advantage of the fact that