In this last lecture of this unit, we continue with some of
our earlier themes, and then introduce one new notion, the
notion of independence of random variables.
We will start by elaborating a bit more on the subject of
conditional probability mass functions.
We have already discussed the case where we condition a
random variable on an event.
Here we will talk about conditioning a random variable
on another random variable, and we will develop yet
another version of the total probability and total
expectation theorems.
There are no new concepts here, just new notation.
I should say, however, that notation is important, because
it guides you on how to think about problems in the most
economical way.
The one new concept that we will introduce is the notion
of independence of random variables.
It is actually not an entirely new concept.
It is defined more or less the same way as independence of
events, and has a similar intuitive interpretation.
Two random variables are independent if information
about the value of one of them does not change your model or
beliefs about the other.
On the mathematical side, we will see that independence
leads to some additional nice properties
of means and variances.
We will conclude this lecture and this unit on discrete
random variables by considering a rather difficult
problem, the hat problem.
We will see that by being systematic and using some of
the tricks that we have learned, we can calculate the
mean and variance of a rather complicated random variable.