Jacobian  (determinant)

The theory of continuous probability distributions makes heavy use of variable transformations (see here). The most general problem is then :

    * Suppose we have n random variables {X₁, ..., X_n}, with joint probability density function f(x₁,..., x_n).

    * Let also T be a one-to-one transformation (y₁,..., y_n) = T(x₁,..., x_n). This transformation defines n new random variables {(Y₁, ..., Y_n}. What is the joint probability density g(y₁,..., y_n) of {Y₁, ..., Y_n} ?

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The answer is as follows.

Let T ^-1 denote the inverse transformation. T ^-1 is a set of n functions j_i :

    * x_i = j_i (y₁,..., y_n)

Then :

g(y₁,..., y_n) = f( j₁(y₁,..., y_n), ..., j_n(y₁,..., y_n)).|det J|^-1

with :

This somewhat intimidating expression is called the jacobian (determinant) of the transformation T.

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In this Tutorial, we consider the case n = 2, and establish the above expression for g(y₁, y₂). So we do not assume the above formula known, but instead we demonstrate it from first principles and in the process, introduce the notion of "jacobian" in a very natural way.

We'll see that the jacobian receives a straightforward geometric interpretation that can be conveniently illustrated.

The demonstration calls on results in elementary planar geometry, and cannot be directly generalized to higher dimensions. Yet, the mathematical theory of jacobians shows that the geometric interpretation of the jacobian in the 2-D case extends straightforwardly to any dimension.

JACOBIAN DETERMINANT (2-D case)

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