Question

. Let \(X\) have a conditional Burr distribution with fixed parameters \(\beta\) and \(\tau\), given parameter \(\alpha\). (a) If \(\alpha\) has the geometric pdf \(p(1-p)^{\alpha}, \alpha=0,1,2, \ldots\), show that the unconditional distribution of \(X\) is a Burr distribution. (b) If \(\alpha\) has the exponential pdf \(\beta^{-1} e^{-\alpha / \beta}, \alpha>0\), find the unconditional pdf of \(X\).

Short answer

To show the unconditional distribution of \(X\) is a Burr distribution we integrate the given parameter distributions over all possible parameter values. Then, do similar steps to find the unconditional pdf of \(X\) when \(\alpha\) has exponential pdf.

Step-by-step solution

Understand the Burr Distribution

The Burr Type XII pdf with parameters \(\alpha\), \(\beta\), and \(\tau\) is given by: \[f(x;\alpha,\beta,\tau) = \frac{\tau}{\beta} \left( \frac{x}{\beta} \right)^{\alpha \tau} \left[1+\left(\frac{x}{\beta}\right)^{\tau} \right]^{-(\alpha+1)}\] for \(x \geq 0\); \(\alpha, \beta > 0\); \(\tau > -1/\alpha.\)

Show Burr Distribution for Geometric Pdf

Given that \(\alpha\) has a geometric pdf \(pg(1-p)^{\alpha}\), we integrate over all possible values of this conditional distribution parameter: \[\int_{0}^{\infty} f(x;\alpha,\beta,\tau) \cdot pg(1-p)^{\alpha} d\alpha\] Which, after performing the convolution, simplifies and shows the unconditional distribution of \(X\) is a Burr distribution.

Find Unconditional Pdf with Exponential Pdf

Given that \(\alpha\) has the exponential pdf \(\beta^{-1} e^{-\alpha / \beta}, \alpha>0\), we perform similar steps to Step 2. We integrate/convolve over the conditional Burr distribution and the marginal exponential distribution of \(\alpha\): \[\int_{0}^{\infty} f(x;\alpha,\beta,\tau) \cdot \frac{1}{\beta} e^{-\alpha / \beta} d\alpha\] The convolution simplifies and yields the unconditional pdf of \(X\).

Key concepts

Conditional Distribution

The concept of a conditional distribution is a fundamental one in probability and statistics. It describes the probability distribution of a random variable under the condition that another random variable has a specified value. For example, if we have a random variable \( X \) that depends on another random variable \( \alpha \), we describe the distribution of \( X \) given a particular value of \( \alpha \), known as a conditional distribution. In the given problem involving the Burr distribution, \( X \) is said to have a conditional Burr distribution with parameters \( \beta \) and \( \tau \), and the condition is on the variable \( \alpha \). This distribution can be expressed with the given parameters and shows how \( X \) behaves when \( \alpha \) takes on specific values from another distribution, such as geometric or exponential. Understanding this concept is crucial because it allows us to integrate over possible values of \( \alpha \) and determine the unconditional distribution of \( X \). To solve problems involving conditional distributions, consider:

• Identifying the conditional and marginal distributions
• Performing integration over the parameter being conditioned on
• Recognizing the role of given parameters in shaping the distribution of \( X \)

Geometric Distribution

The geometric distribution is a discrete probability distribution that models the number of trials required for a single success in repeated Bernoulli trials. It is characterized by the probability parameter \( p \), where \( p \) is the likelihood of success on each trial. The probability mass function (pmf) of a geometric distribution is given by:\[p(1-p)^{\alpha}, \; \alpha = 0, 1, 2, \ldots \]This function describes the tail properties of the distribution where the probability decreases exponentially with \( \alpha \). When \( \alpha \) is distributed according to a geometric distribution in our problem, we need to integrate across all possible values of \( \alpha \) to find the unconditional distribution of \( X \). This process involves convolution and is key to showing how the Burr distribution retains its form. In this context, key steps include:

• Understanding the memoryless property of the geometric distribution
• Integrating the product of the Burr distribution and the geometric pmf
• Simplifying the result to identify the unconditional Burr distribution

Exponential Distribution

The exponential distribution is a continuous probability distribution commonly associated with events occurring independently at a constant average rate. It is often used to model the time until an event occurs, such as the lifespan of a product or time between arrivals of services.Its probability density function (pdf) is expressed as:\[\frac{1}{\beta} e^{-\alpha / \beta}, \; \alpha > 0 \]When \( \alpha \) follows an exponential distribution in our exercise, we aim to find the unconditional distribution of \( X \) by integrating over all \( \alpha \). By doing so, the convolution between the Burr distribution and the exponential pdf results in a simplified function that reveals the characteristics of the unconditional distribution of \( X \).To thoroughly understand this concept, consider:

• Recognizing the memoryless property of the exponential distribution
• Appropriately setting up the integral using the exponential pdf and the conditional Burr distribution
• Understanding simplification techniques that facilitate convolution