Transformation

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 $\mathbf{X}=(X_1,\dots ,X_n)$ into $\mathbf{Y}=(Y_1,\dots ,Y_n)$ and wanting to find the joint PDF of $\mathbf{Y}$, given $\mathbf{X}$‘s. An example is rectangular to polar coordinates.
• Given that $X_1,\dots ,X_n$ 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 $\text{Support}(X)$.

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

$$f_Y(y)=f_X(x)\cdot \left|\frac{dx}{dy}\right|$$

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 $dx/dy=\frac{1}{dy/dx}$, so we can calculate whichever one is easier.

• The support of $Y$ is given by all $g(x)$ for $x\in \text{Support}(X)$.