In the previous lecture we introduced random variables,
probability mass functions and expectations.
In this lecture we continue with the development of
various concepts associated with random variables.
There will be three main parts.
In the first part we define the variance of a random
variable, and calculate it for some of our
familiar random variables.
Basically the variance is a quantity that measures the
amount of spread, or the dispersion of a probability
mass functions.
In some sense, it quantifies the amount of randomness that
is present.
Together with the expected value, the variance summarizes
crisply some of the qualitative properties of the
probability mass function.
In the second part we discuss conditioning.
Every probabilistic concept or result has a conditional
counterpart.
And this is true for probability mass functions,
expectations and variances.
We define these conditional counterparts and then develop
the total expectation theorem.
This is a powerful tool that extends our familiar total
probability theorem and allows us to divide and conquer when
we calculate expectations.
We then take the opportunity to dive deeper into the
properties of geometric random variables, and use a trick
based on the total expectation theorem to
calculate their mean.
In the last part we show how to describe probabilistically
the relation between multiple random variables.
This is done through a so-called joint probability
mass function.
We take the occasion to generalize the expected value
rule, and establish a further linearity property of
expectations.
We finally illustrate the power of these tools through
the calculation of the expected value of a binomial
random variable.