Interactive element. The change of variables describes how the distribution p(x) associated with a random variable x changes under a function f that yields a new random variable y = f(x). Drag the nodes to adjust the cubic spline and see how the pushforward distribution (blue) changes in response to the function.

## Change of variables

The change of variables formula describes what happens to a pdf p(x) when its associated random variable x is transformed with a function f, which yields a new random variable y = f(x) associated with a pushforward pdf p(y):

p(f(x)) = p(x) / |∂ f(x)|

In words: the probability density p(y) after the transformation depends on the probability density before the transformation (p(x)) corrected for any stretching or squeezing induced by the transformation (1 / |∂ f(x)| ).

Observe that the steeper the gradient of f is, the smaller 1 / |∂ f(x)| and the lower the pushforward density becomes. Conversely, if the gradient of f is flat, the correction term inflates the density.

Drag your mouse over the graph and observe how the function stretches or squeezes the red input interval.