In this lecture, we continue our discussion of continuous
random variables.
We will start by bringing conditioning into the picture
and discussing how the PDF of a continuous random variable
changes when we are told that a certain event has occurred.
We will take the occasion to develop counterparts of some
of the tools that we developed in the discrete case such as
the total probability and total expectation theorems.
In fact, we will push the analogy even further.
In the discrete case, we looked at the geometric PMF in
some detail and recognized an important memorylessness
property that it possesses.
In the continuous case, there is an entirely analogous story
that we will follow, this time involving the exponential
distribution which has a similar
memorylessness property.
We will then move to a second theme which is how to describe
the joint distribution of multiple random variables.
We did this in the discrete case by
introducing joint PMFs.
In the continuous case, we can do the same using
appropriately defined joint PDFs and by
replacing sums by integrals.
As usual, we will illustrate the various concepts through
some simple examples and also take the opportunity to
introduce some additional concepts such as mixed random
variables and the joint cumulative
distribution function.