Question

Let \(X_{1}, \ldots, X_{n}\) be independent exponential random variables each having rate 1 . Set $$ \begin{aligned} &W_{1}=X_{1} / n \\ &W_{i}=W_{i-1}+\frac{X_{i}}{n-i+1}, \quad i=2, \ldots, n \end{aligned} $$ Explain why \(W_{1}, \ldots, W_{n}\) has the same joint distribution as the order statistics of a sample of \(n\) exponentials each having rate 1 .

Short answer

In order to show that \(W_{1}, \ldots, W_{n}\) have the same joint distribution as the order statistics of a sample of n exponentials each having rate 1, we first find the joint density function of the independent exponentials. Next, we compute the Jacobian of the transformation mapping the random variables \(X_1, \dots, X_n\) to \(W_1, \dots, W_n\). Using the change of variables theorem, we find the joint density function of the transformed random variables and observe that it has the same form as the joint distribution of the order statistics of n exponentials each having rate 1, which implies that the given random variables share the same joint distribution.

Step-by-step solution

Background information on order statistics and joint distributions

An order statistic of a given sample is simply a statistic defined on the ordered values of the sample. For a given exponential distribution, with rate 1, we usually consider the order statistics to be the times of the'n' arrivals, which are exponentially distributed. By examining the joint distribution, we can observe the relationship between the probabilities of different outcomes or events in multi-dimensional space.

Write down the joint density function of independent exponentials

Since \(X_{1}, \ldots, X_{n}\) are independent exponential random variables, each having rate 1, their joint density function can be represented as follows: \(f_{X}(x_{1}, \ldots, x_{n}) = \prod_{i=1}^{n} e^{-x_{i}} = e^{-\sum_{i=1}^{n} x_{i}},\) for \(x_i \geq 0\) for all \(i\).

Find the Jacobian of the transformation

In order to find the joint distribution of \(W_{1}, \ldots, W_{n}\), we need to perform a transformation that maps the random variables \(X_1, \dots, X_n\) to \(W_1, \dots, W_n\). To do this, we first find the Jacobian of the transformation. Let's write down the transformation functions: \[ \begin{aligned} &W_{1}=X_{1} / n \\ &W_{i}=W_{i-1}+\frac{X_{i}}{n-i+1}, \quad i=2, \ldots, n \end{aligned} \] Now, compute the Jacobian matrix, J: \[ J = \begin{bmatrix} \frac{\partial W_{1}}{\partial X_{1}} & \frac{\partial W_{1}}{\partial X_{2}} & \cdots & \frac{\partial W_{1}}{\partial X_{n}} \\ \\ \frac{\partial W_{2}}{\partial X_{1}} & \frac{\partial W_{2}}{\partial X_{2}} & \cdots & \frac{\partial W_{2}}{\partial X_{n}} \\ \\ \vdots & \vdots & \ddots & \vdots \\ \\ \frac{\partial W_{n}}{\partial X_{1}} & \frac{\partial W_{n}}{\partial X_{2}} & \cdots & \frac{\partial W_{n}}{\partial X_{n}} \\ \end{bmatrix} \] We can calculate the entries of this matrix: \[ \begin{aligned} & \qquad \frac{\partial W_{1}}{\partial X_{i}} = 0, \quad i \neq 1 \\ & \qquad \frac{\partial W_{1}}{\partial X_{1}} = \frac{1}{n} \\ & \qquad \frac{\partial W_{i}}{\partial X_{j}} = 0, \quad j \neq i \\ & \qquad \frac{\partial W_{i}}{\partial X_{i}} = \frac{1}{n-i+1}, \quad i = 2, \ldots, n \end{aligned} \] Now, compute the determinant of J, denoted by |J|: \[ \begin{aligned} |J| = \frac{1}{n} \times \frac{1}{n-1} \times \cdots \times \frac{1}{1} = \frac{1}{n!} \end{aligned} \]

Use the joint density function and change of variables theorem

By applying the change of variables theorem, we can find the joint density function of the transformed random variables \(W_1, \dots, W_n\): \( \begin{aligned} f_{W}(w_{1}, \ldots, w_{n}) = f_{X}(x_{1}(w), \ldots, x_{n}(w)) \times |J| \end{aligned} \) By substituting the joint density function of independent exponentials and the Jacobian, we get: \( \begin{aligned} f_{W}(w_{1}, \ldots, w_{n}) = e^{-\sum_{i=1}^{n} x_{i}(w)} \frac{1}{n!} \end{aligned} \)

Observe the equivalence of the joint distributions

By comparing the obtained joint density function to the joint distribution of the order statistics of n exponentials each having rate 1, which is \(g(w_1, w_2, ..., w_n) = n! e^{-n w_1 - (n-1)w_2 -...- w_n} I_{(0 \leq w_1 \leq w_2 \leq ... \leq w_n)}\), we can clearly see that both joint distribution functions have the same form: \( \begin{aligned} f_{W}(w_{1}, \ldots, w_{n}) = n! \times g(w_1, w_2, \ldots, w_n) \end{aligned} \) This equivalence in the distributions shows that the random variables \(W_{1}, \ldots, W_{n}\) have the same joint distribution as the order statistics of a sample of n exponentials each having rate 1.

Key concepts

Understanding Exponential Random Variables

In the realm of probability and statistics, exponential random variables are a key concept to grasp, especially when dealing with time until an event occurs such as the time it takes for a radioactive atom to decay or the time between customer arrivals in a queue.

An exponential random variable is defined by its rate parameter (often denoted by \( \lambda \)) which describes the average number of events in a given time interval. For a rate of 1, as in our exercise, the probability density function (PDF) is given by \( f(x) = e^{-x} \), for \( x \geq 0 \). The key properties that make exponential random variables unique include the memoryless property, which indicates that the passage of time does not affect the probability of the event occurring, and their role in modeling the time between independent events that occur at a constant average rate.

Within the context of the provided exercise, exponential random variables represent the times between events in a sequence of independently occurring events. By analyzing the behavior of these variables, particularly through their order statistics, one can learn about the underlying distribution of waiting times or arrival times.

Joint Distribution of Random Variables

In a multidimensional stochastic environment, the joint distribution becomes a necessary tool to understand how multiple random variables interact with one another. When we talk about joint distribution, we focus on the probability that each of the variables falls within a particular range or set of values simultaneously.

For independent exponential random variables with rate 1, we can describe the joint distribution as the product of their individual PDFs, thanks to their independence. The resulting joint density function is expressed as \( f_{X}(x_{1}, \ldots, x_{n}) = e^{-\sum_{i=1}^{n} x_{i}} \), for \(x_i \geq 0\) for all \(i\). This shows how the probability of observing a particular vector of values is the product of the probabilities of observing each value separately.

In the exercise example, understanding the concept of joint distribution is crucial for demonstrating that the sequence of transformed variables \( W_{1}, \ldots, W_{n} \) has the same joint distribution as the order statistics from a sample of exponential variables, which is a fundamental step in proving the desired equivalence.

Transformation of Variables

The concept of transformation of variables in probability theory allows us to convert or map a set of random variables into another set through a specified function. This transformation impacts how we understand the behavior of the variables, and in certain cases, can simplify complex problems.

A critical component of this technique is determining the Jacobian matrix, which contains all the first-order partial derivatives of the transformation functions. The determinant of the Jacobian matrix, commonly denoted as \( |J| \), provides a factor that scales or adjusts the probability densities as a result of the transformation.

In the exercise, the provided transformation functions restructure the original exponential random variables into a new set of variables that correspond to their order statistics. Computing the Jacobian determinant as \( \frac{1}{n!} \) and applying the change of variables theorem allows us to derive the joint density function for the transformed variables. This ultimately demonstrates that the random variables \( W_{1}, \ldots, W_{n} \) after transformation share the same joint distribution characteristics as the order statistics from the original set of exponentials. Understanding this transformation is essential for students to appreciate how probability distributions behave under manipulation and mapping of random variables.