Problem 23 - Calculus Early Transcendentals: Pearson New International Edition

Question

Suppose \(X\) and \(Y\) are continuous random variables with joint 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 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.

Step-by-Step Solution

Answer

The joint PDF of \(U\) and \(V\) is \(g(u, v) = f(x(u, v), y(u, v))|J(u, v)|\), derived using the change of variables and Jacobian determinant in probability integrals.

Step 1: Define Joint PDF transformation

Given the transformation from random variables \(X, Y\) to \(U, V\), we are tasked with finding the joint PDF of \(U\) and \(V\). This requires expressing the probability distribution function of \(X\) and \(Y\) in terms of \(U\) and \(V\).

Step 2: Use probability equivalence

We need to show that \(P((X, Y) \in R) = P((U, V) \in S)\) for regions \(R\) and \(S\). Since \(R\) in the \(xy\)-plane maps one-to-one to \(S\) in the \(uv\)-plane, these probabilities are equivalent by the properties of continuous random variables.

Step 3: Set up double integrals for probabilities

For \(P((X, Y) \in R)\), we have a double integral over region \(R\): \[ P((X, Y) \in R) = \int\int_{R} f(x, y) \, dx \, dy \]For \(P((U, V) \in S)\), we have:\[ P((U, V) \in S) = \int\int_{S} g(u, v) \, du \, dv \]

Step 4: Relate the integrals through transformation

Using the transformation \((x, y) \rightarrow (u, v)\), replace \(dx\) and \(dy\) with \(du\) and \(dv\) in the integral using the Jacobian determinant. The Jacobian matrix \(J\) of the transformation has entries \[ J = \begin{bmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{bmatrix} \] and the determinant \(|J(u, v)|\) is used as the scaling factor.

Step 5: Write the joint PDF using transformation rule

Substitute in the relation:\[ g(u, v) = f(x(u, v), y(u, v))\cdot |J(u, v)| \]The formula is derived from the change of variables in double integrals, factoring in the Jacobian determinant.

Step 6: Finalize and interpret results

Thus, the joint PDF of \(U\) and \(V\), \(g(u, v) = f(x(u, v), y(u, v))|J(u, v)|\), accounts for the deformation of area using the Jacobian in this transformation, preserving the equivalence of probabilities in corresponding regions.

Key Concepts

Continuous Random Variables

Continuous random variables are fundamental in probability theory because they can take on an infinite number of values within a given range. Unlike discrete random variables, which are restricted to specific values, continuous variables can assume any value within a certain interval. This characteristic makes continuous random variables indispensable when modeling real-world phenomena such as weights, heights, or times.To describe a continuous random variable, we use a probability density function (PDF), denoted as \(f(x)\) for a random variable \(X\). This function helps in calculating the probability that the variable falls within a particular range. Specifically, the probability that the variable lies between two values \(a\) and \(b\) is given by the integral of the PDF over this interval:\[P(a \leq X \leq b) = \int_{a}^{b} f(x) \, dx\]Thus, understanding continuous random variables is essential for grasping more complex concepts like joint probability density functions, especially when multiple continuous random variables are involved in an analysis.

Transformation of Variables

Transformation of variables is a powerful technique used to simplify the analysis of random variables and their joint distributions. By converting one set of variables to another, we can often find more convenient ways to express and solve probability functions.In our case, transforming variables plays a crucial role in understanding how a joint probability distribution of two variables \((X, Y)\) changes when we switch to a new pair \((U, V)\). This transformation is given by the functions \(U = g(X, Y)\) and \(V = h(X, Y)\). The transformation is specifically useful when it's one-to-one, which means that each point in the \((X, Y)\) space corresponds uniquely to a point in the \((U, V)\) space. This one-to-one relationship ensures that probabilities in the \((X, Y)\) space are equivalently represented in the \((U, V)\) space, making transformations a key tool for simplifying complex probabilistic functions.

Change of Variables in Integrals

Change of variables is a method often used in calculus to evaluate complex integrals by transforming the variables of integration. When dealing with probability, especially in the context of continuous random variables, this becomes extremely valuable.In probability, the concept is extended to double integrals involving joint PDFs. Consider transforming variables from \((X, Y)\) to \((U, V)\). The probability that \((X, Y)\) lies in a region \(R\) can be expressed as a double integral over that region's joint PDF:\[P((X, Y) \in R) = \int \int_{R} f(x, y) \, dx \, dy\]To express this in terms of \((U, V)\), a change of variables is applied. This involves substituting \(x(u, v)\) and \(y(u, v)\) into the joint PDF and adjusting the differential \(dx \, dy\) to \(du \, dv\) using the Jacobian determinant. This technique helps maintain the probability measure under the transformation, ensuring the double integrals in the original and transformed space remain equivalent.

Jacobian Determinant

The Jacobian determinant is a central concept in the transformation of variables in multivariable calculus. It provides a measure of how much a function stretches, shrinks, or otherwise distorts space around a given point during a transformation.When transforming variables in a double integral, the Jacobian determinant is the factor that adjusts the differential area element to maintain the total probability under the transformation. For our transformation from \((X, Y)\) to \((U, V)\), the Jacobian matrix \(J\) is represented as:\[J = \begin{bmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{bmatrix}\]The absolute value of its determinant \(|J(u, v)|\) is then used as a scaling factor in the joint PDF of \(U\) and \(V\):\[g(u, v) = f(x(u, v), y(u, v)) \cdot |J(u, v)|\]This ensures that the transformation accurately reflects the new variable space while preserving the overall probability measure. Understanding and calculating the Jacobian determinant is essential for any transformation involving continuous random variables.