Consider a probability density $p_x(x)$ defined over a continuous variable x, and suppose that we make a nonlinear change of variable using $x = g(y)$. The location $\hat{y}$ of the maximum of the density in $y$ is not in general related to the location $\hat{x}$ of the maximum of the density over x by the simple functional relation $\hat{x} = g(\hat{y})$.
Relevant Theorem: The Change of Variable Theorem, also known as the Transformation Theorem, is a fundamental concept in probability theory that deals with the transformation of random variables. It allows us to find the probability distribution of a new random variable that is derived from an existing random variable through a deterministic function.

Suppose we have a random variable X with a known probability density function (pdf) f_X(x) and a cumulative distribution function (cdf) F_X(x). Let Y = g(X) be a new random variable obtained by applying a function g(x) to X, where g(x) is a continuous and differentiable function with an inverse function g^(-1)(y).

The Change of Variable Theorem states that the pdf of the transformed random variable Y, denoted as f_Y(y), can be obtained using the following formula:

f_Y(y) = f_X(x) * |(dg^(-1)(y) / dy)|

where x = g^(-1)(y) and |(dg^(-1)(y) / dy)| is the absolute value of the derivative of the inverse function g^(-1)(y) with respect to y.

The theorem essentially provides a method to compute the pdf of the transformed random variable Y by considering the pdf of the original random variable X and the transformation function g(x). This is particularly useful in various applications, such as statistical modeling and hypothesis testing, where we often need to work with transformed random variables.
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