Question

Let \(X\) and \(Y\) denote independent random variables with respective probability density functions \(f(x)=2 x, 0

Short answer

The joint pdf \(f_{U,V}(u,v) = 2u . 3v^{2} + 2v . 3u^{2}\) for \(0 < u < v < 1\)

Step-by-step solution

Note the given transformations

We're given the transformations \(x=u, y=v\) and \(x=v, y=u\). This indicates how U and V are related with X and Y. Note that U and V are both functions of X and Y.

Generating Jacobian of the transformations

We need to calculate the Jacobian determinant of the transformation to continue with the problem. The Jacobian \(J\) of a transformation is the determinant of the matrix of partial derivatives, which in our case will be \(J = |dx/du, dy/du, dx/dv, dy/dv|\). For the given transformations, the Jacobian will be \(|1, 0, 0, 1|\) and \(|0, 1, 1, 0|\) for each transformation, and both of them equal to 1.

Calculate Probability

Now that we have the Jacobian, we can use that to get the joint density of U (minimum of X and Y) and V (maximum of X and Y). Our joint density function will be the sum of the joint densities of (u, v) for each transformation, multiplied by the absolute value of the Jacobian (which we've calculated as 1). Based on the transformations and the individual pdfs of X and Y given, we have two cases: 1) \(f_{U,V}(u,v) = |J| . f_X(u) . g_Y(v)\) for \(0 < u < v < 1\), 2) \(f_{U,V}(u,v) = |J| . f_X(v) . g_Y(u)\) for \(0 < u < v < 1\). Now we calculate each of those densities, sum them to acquire the joint pdf of U and V.

Key concepts

Probability Density Functions

Probability density functions (pdfs) are fundamental to understanding random variables in probability and statistics. They characterize the distribution of continuous random variables, illustrating how the probabilities are distributed over the values the random variables can take.

Pdfs must satisfy two conditions: firstly, the probability density function must be non-negative for all possible values of the variable; secondly, the area under the pdf across the entire range must equal 1, signifying the total probability. In our exercise, the pdfs for the random variables X and Y are given by specific mathematical functions over the interval from 0 to 1, with values outside this range being zero.

When dealing with joint probability density functions for multiple variables, such as in the case of finding the pdf for the minimum (U) and the maximum (V) of X and Y, it's imperative to understand how the pdfs of individual variables combine to determine the probabilities associated with the pair (U, V).

Jacobian Determinant

The Jacobian determinant is a critical tool when transitioning between variables in multivariable calculus, particularly when transforming probability density functions. It's defined as the determinant of a matrix of first-order partial derivatives and measures how a multivariate function transforms areas in the space of its input variables.

In the context of the exercise, the Jacobian is essential when finding the joint pdf of new variables U and V, derived from the original variables X and Y. Since U and V are functions of X and Y, we need the Jacobian to adjust the scale appropriately and ensure probabilities are conserved in the transformation. If the Jacobian is positive, it indicates the region's orientation is preserved, while a negative Jacobian would indicate a flip in orientation.

Fortunately, in our example, the Jacobian determinant is 1 for both transformations. This simplifies the process, as there's no correction factor needed when finding the joint pdf for U and V.

Random Variables

Random variables are foundational in probability theory, representing possible outcomes of a random phenomenon numerically. A random variable is essentially a function that assigns numerical values to each outcome of an experiment.

In this exercise, X and Y are independent random variables, which means that the occurrence of one does not affect the probability distribution of the other. Independence is a powerful concept because it allows the joint pdf of X and Y to be expressed as the product of their individual pdfs. However, when defining new random variables in terms of existing ones—like U as the minimum of X and Y, and V as their maximum—we often introduce dependence between these new variables. We then need to use techniques, including but not limited to the Jacobian determinant, to correctly describe their joint distribution.