We now note some elementary properties of expectations.
These will be some properties that are extremely natural and
intuitive, but even so, they are worth recording.
The first property is the following.
If you have a random variable which is non-negative, then
its expected value is also non-negative.
What does it mean that the random variable is
non-negative?
What it means is that for all possible outcomes of the
experiment, no matter what the outcome is, the associated
numerical value of the random variable is a
non-negative number.
What's the implication of this?
When we calculate an expectation we're adding over
all the possible numerical values of the random variable.
All the possible numerical values of the random variable
under this assumption are non-negative.
Probabilities are also non-negative.
So we have a sum of non-negative entries and
therefore, the expected value is also going to be
non-negative.
The next property is a generalization of this.
Consider now a random variable that has the property that no
matter what the outcome of the experiment is, the value of
this random variable lies in the range between two
constants, a and b.
In this case, we argue as follows.
The expected value, by definition, is a sum over all
possible values of the random variable of certain terms.
Now, the possible numerical values of the random variable
are all of them at least as large as a, so this gives us
an inequality of this type.
Then, we pull a factor of a outside of the summation.
And finally, we recall that the sum of a PMF over all
possible values of little x is equal to 1.
Why is that the case?
Well, these are the probabilities for the
different numerical values of the random variable.
The sum of the probabilities of all the possible numerical
values has to be equal to 1, because that exhausts all the
possibilities.
So we obtain a times 1, which is a.
So, what we have proved is that the expected value is at
least large as a.
You can use a symmetrical argument where the
inequalities will go the opposite way and where a's
will be replaced by b's, to prove the second
inequality, as well.
The last fact we want to take note of is the following.
If we have a constant and we take its expected value, we
obtain the same constant.
What does that mean?
We have only been talking about expected values of
random variables.
What does it mean to take the expected value of a constant?
Well, as we discussed earlier, we can think of a constant as
being a random variable of a very special type.
A random variable whose PMF takes this form.
This random variable can take only a single value and the
probability of that single value is equal to 1.
This means that in the formula for the expected value there's
going to be only one term in this summation, and that term
is going to be c times the probability that our random
variable takes the value c.
Now, that probability is equal to 1, and we're left with c.
So this equality makes sense, of course, as long as you
understand that a constant can also be viewed as a random
variable of a very degenerate type.
Now, intuitively, of course, it's certainly clear
what this is saying.
That if a certain quantity is always equal to c, then on the
average, it will also be equal to c.