Question

Suppose \(X\) and \(Y\) are continuous random variables with joint \(\operatorname{PDF} f(x, y)\) and suppose \(U\) and \(V\) are random variables that are functions of \(X\) and \(Y\) such that the transformation $$ X=x(U, V) \quad \text { and } \quad Y=y(U, V) $$ is one-to-one. Show that the joint \(\mathrm{PDF}\) of \(U\) and \(V\) is $$ g(u, v)=f(x(u, v), y(u, v))|J(u, v)| $$ Hint: Let \(R\) be a region in the \(x y\) -plane and let \(S\) be its preimage. Show that \(P((X, Y) \in R)=P((U, V) \in S)\) and get a double integral for each of these.

Short answer

The joint PDF of \(U\) and \(V\) is given by \(g(u, v) = f(x(u, v), y(u, v)) |J(u, v)|\).

Step-by-step solution

Understand the Setup

We are given a transformation of random variables \(X\) and \(Y\) into new variables \(U\) and \(V\) using a one-to-one function. The goal is to find the joint probability density function (PDF) of \(U\) and \(V\). The transformation involves functions \(x(u, v)\) and \(y(u, v)\) which map \(U\) and \(V\) back to \(X\) and \(Y\).

Express Probabilities using PDFs

The probability that \((X, Y)\) falls into a region \(R\) is expressed as an integral: \[ P((X, Y) \in R) = \iint_R f(x, y) \, dx \, dy. \]Similarly, the probability that \((U, V)\) falls into a preimage \(S\) is:\[ P((U, V) \in S) = \iint_S g(u, v) \, du \, dv. \]

Use the Transformation

Since the transformation \((U, V)\) to \((X, Y)\) is one-to-one, the change of variables for integrals can be applied. We have:\[ dx \, dy = |J(u, v)| \, du \, dv, \]where \(J(u, v)\) is the Jacobian determinant of the transformation from \((u, v)\) to \((x, y)\).

Equate the Two Double Integrals

By equating the probabilities from Step 2, we have:\[ \iint_R f(x, y) \, dx \, dy = \iint_S g(u, v) \, du \, dv. \]Substituting the change of variables, we get:\[ \iint_S f(x(u, v), y(u, v)) \, |J(u, v)| \, du \, dv = \iint_S g(u, v) \, du \, dv. \]

Conclude that the PDF of (U, V) Matches the Transformed PDF

Since the regions were arbitrary, we can conclude that the integrands must be equal for all \(u, v\). Thus, we have:\[ g(u, v) = f(x(u, v), y(u, v)) \, |J(u, v)|. \] This shows that the joint PDF of \(U, V\) is the original joint PDF transformed by the Jacobian.

Key concepts

Continuous Random Variables

In probability theory, continuous random variables are variables that can take on an infinite number of values. These values aren't just fixed numbers; instead, they form a continuum on the real number line. For instance, measurements like height, temperature, and time can all be considered continuous since they can take on any value within certain limits.

Continuous random variables are characterized by probability density functions (PDFs). A PDF doesn't give probabilities directly but describes the density of probability across different outcomes. The probability that a continuous random variable falls within a certain range is given by the integral of its PDF over that range. This integral is essentially the area under the curve described by the PDF over the specified interval.

• This concept is fundamental because for any region in the plane defined by two continuous variables, the overall probability is determined through double integration.
• The joint PDF of two variables provides the density across an area on this plane instead of a single dimension.

Change of Variables

The change of variables is a powerful technique used to transform a set of variables. It allows solving problems more easily by shifting to a more convenient coordinate system. In context, transforming the random variables \(X\) and \(Y\) to new variables \(U\) and \(V\) through defined functional relationships is often required for computing probabilities over different regions.

Mathematically, this is represented as \(X = x(U, V)\) and \(Y = y(U, V)\). Such transformations are especially useful in complex integrations where direct computation is difficult.

• Using a one-to-one transformation ensures that each pair of values in the domain corresponds to exactly one pair in the codomain, preserving properties across conversion.
• The regions in which we integrate, \(R\) and its preimage \(S\), have corresponding probability representations in the new variable space as well.

Jacobian Determinant

The Jacobian determinant plays a crucial role in the transformation of variables, especially in multiple integrations. When you transform variables in a multivariable function, you need to account for how areas (and probabilities) change under that transformation. This is where the Jacobian determinant \(|J(u, v)|\) comes in.

The Jacobian is a matrix that contains all first-order partial derivatives of a function. For functions \(X = x(U, V)\) and \(Y = y(U, V)\), the Jacobian determinant is defined as:\[|J(u, v)| = \left| \begin{array}{cc} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{array} \right|\]

• The determinant measures how much the function expands or contracts regions of space under the transformation.
• By multiplying the original PDF by the absolute value of the Jacobian, the transformed PDF correctly reflects these changes.
• In the context of a one-to-one transformation, this becomes especially critical as it ensures the equality of probability measures under transformation.