Problem 55 - A First Course in Probability

Question

\(X\) and \(Y\) have joint density function $$f(x, y)=\frac{1}{x^{2} y^{2}} \quad x \geq 1, y \geq 1$$ (a) Compute the joint density function of \(U=X Y, V=\) \(X / Y.\) (b) What are the marginal densities?

Step-by-Step Solution

Answer

The joint density function of \(U\) and \(V\) is \(f_{U,V}(u,v) = \begin{cases} \frac{v}{u^2} &\text{ for } u\ge1, v\ge1 \\ 0 & \text{ otherwise } \end{cases}\). The marginal densities of \(U\) and \(V\) are: $$f_U(u) = \frac{1}{2u^2}, \;\; u\ge1$$ $$f_V(v) = \begin{cases} v &\text{ for } v\ge1 \\ 0 & \text{ otherwise } \end{cases}$$

Step 1: Compute the Jacobian of the transformation

To compute the joint density function of the new random variables \(U\) and \(V\), we first need to find the Jacobian of the transformation. Define \(g\) as the transformation from \((X,Y)\) to \((U,V)\). $$g(X,Y) = \begin{bmatrix} U \\ V \end{bmatrix} = \begin{bmatrix} XY \\ X/Y \end{bmatrix}.$$ Now let's find the inverse transformation, which we'll call \(h\). $$h(U,V) = \begin{bmatrix} X \\ Y \end{bmatrix} = \begin{bmatrix} UV \\ U/V \end{bmatrix}.$$ Now, we'll compute the Jacobian, defined as $J = \frac{\partial(X,Y)}{\partial(U,V)} = \det\begin{bmatrix} \frac{\partial X}{\partial U} & \frac{\partial X}{\partial V} \\ \frac{\partial Y}{\partial U} & \frac{\partial Y}{\partial V} \end{bmatrix}.$$

Step 2: Find the partial derivatives

Compute the partial derivatives for the Jacobian. $$\frac{\partial X}{\partial U} = V \quad \frac{\partial X}{\partial V} = U$$ $$\frac{\partial Y}{\partial U} = -\frac{Y}{V^2} = -\frac{U}{V^3} \quad \frac{\partial Y}{\partial V} = \frac{Y}{U}$$ Now, we can compute the Jacobian.

Step 3: Compute the Jacobian

Compute the Jacobian using the partial derivatives. $$J = \det\begin{bmatrix} V & U \\ -\frac{U}{V^3} & \frac{Y}{U} \end{bmatrix} = V\left(\frac{Y}{U}\right) - U\left(-\frac{U}{V^3}\right) = V\left(\frac{U}{V}\right) + U\left(\frac{U}{V^3}\right) = \frac{U}{V} + U\left(\frac{1}{V^3}\right) = \frac{U}{V^3}$$

Step 4: Find the joint density function of U and V

Use the Jacobian to compute the joint density function of \(U\) and \(V\). $$f_{U,V}(u,v) = f_{X,Y}(x(u,v),y(u,v))|J| = \frac{1}{x(u,v)^2y(u,v)^2} \left| \frac{u}{v^3} \right| = \frac{1}{(uv)^2(u/v)^2} \left| \frac{u}{v^3} \right| = \frac{v^3}{u^3v^2}|u| = \frac{v}{u^2}$$ So, the joint density function of \(U\) and \(V\) is \(f_{U,V}(u,v) = \begin{cases} \frac{v}{u^2} &\text{ for } u\ge1, v\ge1 \\ 0 & \text{ otherwise } \end{cases}\).

Step 5: Compute the marginal density functions of U and V

To compute the marginal densities of \(U\) and \(V\), integrate the joint density function. First, we'll compute the marginal density of U. $$f_U(u) = \int_{1}^{\infty} f_{U,V}(u,v)dv = \int_{1}^{\infty} \frac{v}{u^2}dv = \frac{v^2}{2u^2}\Big|_1^\infty = \frac{1}{2u^2}$$ Now, we'll compute the marginal density of V. $$f_V(v) = \int_{1}^{\infty} f_{U,V}(u,v)du = \int_{1}^{\infty} \frac{v}{u^2}du = -\frac{v}{u}\Big|_1^\infty = v$$ Hence, the marginal densities of \(U\) and \(V\) are: $$f_U(u) = \frac{1}{2u^2}, \;\; u\ge1$$ $$f_V(v) = \begin{cases} v &\text{ for } v\ge1 \\ 0 & \text{ otherwise } \end{cases}$$

Key Concepts

Jacobian transformation

The Jacobian transformation is a mathematical tool used to change variables in multi-variable functions, and it plays an essential role in understanding joint density functions for transformed variables. When we need to transform one set of random variables, say \((X, Y)\), into another set \((U, V)\), the Jacobian is involved in ensuring the density functions are appropriately scaled.
In our original exercise, the transformation is from \((X, Y)\) to \((U, V)\), where \(U = XY\) and \(V = \frac{X}{Y}\). To determine how the density function changes with this transformation, we calculate the Jacobian, specifically, the determinant of the partial derivative matrix.

• Transformation Function: \(g(X, Y) = \begin{bmatrix} XY \ X/Y \end{bmatrix}\)
• Inverse Transformation: \(h(U, V) = \begin{bmatrix} UV \ U/V \end{bmatrix}\)

The Jacobian is computed by taking the determinant of the matrix of partial derivatives derived from the inverse transformation. It's denoted as \(J\) and calculated as \[J = \det\begin{bmatrix}\frac{\partial X}{\partial U} & \frac{\partial X}{\partial V} \\frac{\partial Y}{\partial U} & \frac{\partial Y}{\partial V} \\end{bmatrix}\]This determinant helps us find the joint density of \(U\) and \(V\) by scaling the original joint density of \(X\) and \(Y\) by the Jacobian's absolute value.
The transformation ensures that probabilities remain consistent even after such a change of variables.

marginal density

Marginal density is crucial when you want to focus on a single random variable out of a joint distribution. It provides the density of one variable while integrating out the effects of the other. In joint probability distributions, like \(f_{U,V}(u,v)\) for the variables \(U\) and \(V\), the marginal density function extracts the probability distribution of one variable irrespective of the others.
In the original exercise, once we have the joint density function for \(f_{U,V}(u,v)\), we move on to calculate the marginal densities for \(U\) and \(V\) by integrating out the other variable:

• For \(f_{U}(u)\), integrate \(f_{U,V}(u,v)\) with respect to \(v\):\[f_{U}(u) = \int_{1}^{\infty} \frac{v}{u^2} \; dv = \frac{1}{2u^2}, \ u \geq 1\]
• For \(f_{V}(v)\), integrate \(f_{U,V}(u,v)\) with respect to \(u\):\[f_{V}(v) = \int_{1}^{\infty} \frac{v}{u^2} \; du = v, \ v \geq 1\]

These marginal densities show how the individual components \(U\) and \(V\) are distributed, simplifying the analysis from dealing with a joint distribution to comprehending each variable's behavior independently.

partial derivatives

Partial derivatives are a fundamental concept in multivariable calculus and are particularly vital for constructing the Jacobian when performing variable transformations. A partial derivative of a function with respect to one variable is its derivative while keeping other variables constant.
During the Jacobian computation, you find the partial derivatives of the transformation equations with respect to the old and new variables:

• Derivatives related to \(X\): - \(\frac{\partial X}{\partial U} = V\) - \(\frac{\partial X}{\partial V} = U\)

These derivatives form the elements of the Jacobian matrix, essential for calculating the joint density properly.

• Derivatives related to \(Y\): - \(\frac{\partial Y}{\partial U} = -\frac{U}{V^3}\) - \(\frac{\partial Y}{\partial V} = \frac{Y}{U}\)

Once the partial derivatives are structured into a matrix and the determinant is calculated, it gives you the Jacobian, which helps in transforming the original joint density function in a correct and consistent manner. This underlines their importance in multivariable calculus and probability.