Engineering Analysis/SISO Transformations

Let's say that we have a random variable X that is the input into a given system. The system output, Y is then also a random variable that is related to the input X by the response of the system. In other words, we can say that:

$Y=g(X)$

Where g is the mathematical relationship between the system input and the system output.

To discover information about Y, we can use the information we know about the r.v. X, and the relationship g:

$f_{Y}(y)=\sum _{i}{\frac {f_{X}(y)}{\left|{\frac {dy}{dx}}\right|}}$

Where x_i are the roots of g.