Question

Let \(X\) have the conditional Weibull pdf. $$ f(x \mid \theta)=\theta \tau x^{\tau-1} e^{-\theta x^{\tau}}, \quad 0

Short answer

To show that the compound marginal pdf of \(X\) is a Burr distribution, express the joint pdf as the product of the conditional Weibull pdf and the gamma pdf, and then calculate the integral of the joint pdf over all \(\theta\) to arrive at the marginal pdf. Evaluating the integral yields the Burr distribution.

Step-by-step solution

Write down the conditional Weibull pdf

The conditional probability density function (pdf) of \(X\) given \(\theta\) is \(f(x \mid \theta)=\theta \tau x^{\tau-1} e^{-\theta x^\tau}\) for \(0 < x < \infty\).

Write down the gamma pdf (weighting function)

The gamma pdf \(g(\theta)\) is given as \(\frac{\beta^\alpha}{\Gamma(\alpha)} \theta^{\alpha-1} e^{-\beta \theta}\) for \(\theta > 0\).

Express the joint pdf

The joint pdf \(h(x, \theta)\) of \(X\) and \(\Theta\) is the product of the conditional Weibull pdf \(f(x \mid \theta)\) and the gamma pdf \(g(\theta)\). So, \(h(x, \theta) = f(x \mid \theta) \times g(\theta) = \theta \tau x^{\tau-1} e^{-\theta x^\tau} \times \frac{\beta^\alpha}{\Gamma(\alpha)} \theta^{\alpha-1} e^{-\beta \theta}\). Simplifying, \(h(x, \theta) = \frac{\beta^\alpha \theta^{\alpha+\tau-2} x^{\tau-1} e^{-(\beta+\theta) x^\tau}}{\Gamma(\alpha)}\).

Compute the marginal pdf

To find the marginal pdf \(f(x)\) of \(X\), you will need to integrate \(h(x, \theta)\) over all values of \(\theta\). The marginal distribution is given by: \(f(x)=\int_0^\infty h(x, \theta) d\theta\). Use this expression to compute the integral: \(f(x)=\int_0^\infty \frac{\beta^\alpha \theta^{\alpha+\tau-2} x^{\tau-1} e^{-(\beta+\theta) x^\tau}}{\Gamma(\alpha)} d\theta\). With the properties of the gamma function, you can evaluate the integral and find that \(f(x)\) follows the Burr distribution.

Key concepts

Understanding Conditional Probability Density Functions

A conditional probability density function (pdf) represents the likelihood of a random variable taking on a certain value given some condition or parameter. In our exercise, this condition is the parameter \theta, which influences the shape and scale of the Weibull distribution.

The given Weibull pdf is conditional on \theta and is expressed mathematically as \(f(x | \theta)=\theta \tau x^{\tau-1} e^{-\theta x^{\tau}}\) for \(0 < x < \text{\theta} \). This specifies how the probability density of \(X\) changes depending on the value of \( \theta\). The ability to understand and work with conditional pdfs is crucial because it allows students to tackle more complex statistical models where parameters can vary, making the analysis more realistic and applicable to real-world scenarios.

Exploring the Gamma Distribution

The gamma distribution is widely used in statistics, often to model the time until an event happens. It is defined by two parameters, \( \alpha \)-- sometimes called the shape parameter, and \( \beta \)-- known as the rate parameter. The pdf for a gamma distribution with these parameters is \(g(\theta) = \frac{\beta^\alpha}{\Gamma(\alpha)} \theta^{\alpha-1} e^{-\beta \theta}\), where \(\Gamma(\alpha)\) represents the gamma function and is defined for \(\theta > 0\).

The significance of the gamma distribution in this exercise stems from its role as the 'weighting function' for \(\theta\), modifying how the conditional Weibull distribution's shape changes with different parameter values. Understanding the properties of the gamma distribution aids in grasping the later steps where it is compounded with the Weibull distribution to yield the Burr distribution.

Understanding the Burr Distribution

The Burr distribution is a flexible probability distribution that can take on many shapes depending on its parameters. It is often used in reliability analysis and life data modeling. The exercise demonstrates how the Burr distribution emerges as a compound or marginal pdf when a Weibull distribution is mixed with a gamma weighting function.

In essence, by integrating the product of the conditional Weibull pdf and the gamma pdf over all values of \(\theta\), what emerges is the marginal pdf of \(X\), which follows a Burr distribution. Recognizing this compound relationship between distributions is essential for applying statistical models to data that exhibit complex behaviors, and it's a powerful tool for predictive analytics.

Marginal Probability Density Functions in Detail

A marginal probability density function provides the probabilities of a random variable in a multivariate distribution without regard to the possible values of other variables. To find the marginal pdf of \(X\), as seen in the solution, you would integrate the joint pdf \(h(x, \theta)\) over all possible values of \(\theta\).

The process of deriving a marginal distribution from a joint distribution is a foundational concept in statistics and informs a multitude of applications. For example, this approach allows practitioners to forecast the probability of events when dealing with several random variables that may not be independent. The concept underscores the importance of understanding how individual probability distributions contribute to the larger, multivariate context.