In this lecture, we introduce the
notion of a random variable.
A random variable is, loosely speaking, a numerical quantity
whose value is determined by the outcome of a probabilistic
experiment.
The weight of a randomly selected
student is one example.
After giving a general definition, we will focus
exclusively on discrete random variables.
These are random variables that take values in finite or
countably infinite sets.
For example, random variables that take
integer values are discrete.
To any discrete random variable we will associate a
probability mass function, which tells us the likelihood
of each possible value of the random variable.
Then we will go over a few examples and introduce some
common types of random variables.
And finally we will introduce a new concept- the expected
value of the random variable, also called the
expectation or mean.
It is a weighted average of the values of the random
variable, weighted according to their respective
probabilities, and has an intuitive interpretation as
the average value we expect to see if we repeat the same
probabilistic experiment independently a
large number of times.
Expected values play a central role in probability theory.
We will look into some of their properties.
And we will also calculate the expected values of the example
random variables that we will have introduced.