In this lecture, we will deal with a single topic.
How to find the distribution, that is, the PMF or PDF of a
random variable that is defined as a function of one
or more other random variables with known distributions.
Why is this useful?
Quite often, we construct a model by first defining some
basic random variables.
These random variables usually have simple distributions and
often they are independent.
But we may be interested in the distribution of some more
complicated random variables that are defined in terms of
our basic random variables.
In this lecture, we will develop systematic methods for
the task at hand.
After going through a warm-up, the case of discrete random
variables, we will see that there is a general, very
systematic 2-step procedure that relies on cumulative
distribution functions.
We will pay special attention to the easier case where we
have a linear function of a single random variable.
We will also see that when the function involved is
monotonic, we can bypass CDFs and jump directly to a formula
that is easy to apply.
We will also see an example involving a function of two
random variables.
In such examples, the calculations may be more
complicated but the basic approach based on CDFs is
really the same.
Let me close with a final comment.
Finding the distribution of the function g of X is indeed
possible, but we should only do it when we really need it.
If all we care about is the expected value of g of X we
can just use the expected value rule.