In this segment we will go very fast through a few
definitions and facts that remain true in
the continuous case.
Everything is completely analogous to
the discrete case.
And there are absolutely no surprises here.
So, for example, we have defined joint PMFs for the
case of more than two discrete random variables.
And we have a bunch of facts about them.
In a similar manor, we can define joint PDFs for more
than two random variables.
And if you have understood the material so far, you can guess
how such a joint PDF will be used.
For example, you can calculate the probability of a three
dimensional set by integrating the joint PDF over that three
dimensional set.
And there are analogs off all of the other formulas that we
have here where we follow the usual recipe.
Sums become integrals, and PMFs are replaced by PDFs.
Finally, when you deal with a random variable, which is
defined as a function of jointly continuous random
variables, we can use an expected value rule that takes
the same form as in the discrete case.
And using the expected value rule, we can establish, once
more, the usual linearity properties of expectations.
So absolutely no surprises here.
The derivations are either completely straightforward.
Or they follow exactly the same line of argument as in
the discrete case, with just minor changes in notation.