We often want to transform random variables and would like to get the same information about the transformed variable as the original.

- Transforming $X$ to $Y$ with $Y=g(X)$. If we know the PDF of $X$, what is the PDF of $Y$? We can have linear transformations (e.g. Celsius to Fahrenheit) or nonlinear (dog to human age, apparently).
- Transforming a random vector $X=(X_{1},…,X_{n})$ into $Y=(Y_{1},…,Y_{n})$ and wanting to find the joint PDF of $Y$, given $X$‘s. An example is rectangular to polar coordinates.
- Given that $X_{1},…,X_{n}$ are independent r.v.s, what is the distribution of its sum or average?

## Change of variables

### One dimension

Suppose $X$ is a continuous r.v. with PDF $f_{X}$. Let $Y=g(X)$, where $g$ is differentiable and either strictly increasing or decreasing on $Support(X)$.

If $g$ does not meet these requirements e.g. $y=x_{2}$ then we have to derive $f_{Y}$ using the CDF of $Y$.

Then the PDF of $Y$ is

$f_{Y}(y)=f_{X}(x)⋅ dydx $Here, we’re taking the derivative of $x$ with respect to $y$, where $x=g_{−1}(y)$, and then the absolute value of that. Remember that $dx/dy=dy/dx1 $, so we can calculate whichever one is easier.

- The support of $Y$ is given by all $g(x)$ for $x∈Support(X)$.