In this lecture, we start our discussion of continuous
random variables.
We will focus on the case of a single continuous random
variable, and we'll describe its distribution using a
so-called probability density function, an object that will
replace the PMFs from the discrete case.
We will then proceed to define the expectation and the
variance of a continuous random variable, and we'll see
that their basic properties remain unchanged.
There will be one new concept--
the cumulative distribution function, which will allow us
to describe, in a unified manner, both discrete and
continuous random variables, even so-called mixed random
variables that have both a discrete and
a continuous component.
In the course of this lecture, we will also introduce some of
the most common continuous random variables--
uniform, exponential, and normal.
We will pay special attention to the normal distribution and
the ways that we can calculate the associated probabilities.