PROFESSOR: The law of total expectation
will give us another important tool for reasoning
about expectations.
And it's basically a rule like the law of total probability,
closely related to it really, for reasoning
by cases about expectation.
So it requires a definition of what's
called conditional expectation.
So the expectation of a random variable R, given event A,
is simply what you get by thinking
of replacing the probability that R equals v
by the probability that R equals v given A.
So it's the sum over all the possible values
that R might take of the probability
that R takes that value, given A.
OK, with that definition, we can state the basic form
of the law of total expectation, which
says if you want to calculate the expectation of R,
you can split it into cases, according to whether
or not A occurs.
It's simply the conditional expectation
of R given A times the probability of A,
plus the conditional expectation of R, given not A times
the probability of not A. So it really
looks [? as ?] the same format as the law
of total probability.
Now, of course it generalizes to many cases.
So the general form would say that I
can calculate the expectation of R
by breaking it up into the case that A 1 holds
times the probability of A 1, the case that A 2 holds
times the probability of A 2, through A n.
And this could very well, and typically is,
an infinite sum, where the [? A i's ?] of course,
are a partition of the sample space--
so they're all the different cases, either A 1 or A 2
or A 3, they're disjoint.
And altogether, they cover the entire set of possibilities.
Well, let's use this to get a nice different and simpler
way-- more elementary way-- of calculating the expected number
of heads and flips.
So let's let of n be the expected number of heads
and flips-- just shorthand, because the notational
will be easier to work with than writing
capital E brackets of H n.
So what do we know about expectation of n?
Well, I can express it in terms of the expectation
of the remaining flips.
So if I have n flips to perform, they're independent.
Then if I perform the first flip, something happens.
And after that I'm going to do n more
flips, and the expected number of flips
is going to be the expected number on the remaining n
minus 1 plus what happened now.
Well, if I flipped a head first, then I've got a 1
as adding to my total number of heads.
And then I'm going to do n more flips,
so the expected number of flips is
going to be that 1 plus the expected
number on the rest of them.
If the first flip was not a head, it was a tail,
then the total expected number of heads
is simply the expected number of heads on the rest of the flips.
And these are two cases where I can apply total expectation.
So by total expectation, the expected number in n flips
is 1 plus e n minus 1 times the probability of a head,
plus e n minus 1 times the probability of a tail.
Well, now we could do a little algebra multiply through here
by p-- that becomes a p, and this becomes a p times
e n and minus 1.
So I've got e n minus 1 times p, and e n minus 1 times
q-- remembering that p plus q is 1, this simplifies to being
simply e n minus 1 plus p.
Well, this is a very simple kind of recursive definition of e n,
because you can see what's going to happen.
Subtracting 1 from n adds a p.
So if I subtract 2 from n, I add another p-- I get 2 p.
And continuing all the way to the end,
by the time I get to 0, I've gotten n times p.
And I've just figured out what I was familiar with already--
which we previously derived by differentiating
the binomial theorem-- the expected number of heads
in n flips is n times p.
But this time I got it in a somewhat more elementary way,
by appealing to total expectation.