Question

Let \(X\) have a conditional Burr distribution with fixed parameters \(\beta\) and \(\tau\), given parameter \(\alpha .\) (a) If \(\alpha\) has the geometric pmf \(p(1-p)^{\alpha}, \alpha=0,1,2, \ldots\), 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

Given the parameters, it's found that the unconditional distribution of \(X\) which is conditional on \(\alpha\) is still a Burr distribution when \(\alpha\) is geometric. The unconditional pdf of \(X\), when \(\alpha\) is distributed exponentially cannot be described exactly without more information on the parameters and the form of the Burr distribution.

Step-by-step solution

Understand Distributions

The Burr distribution is defined by three parameters, often denoted as \(c, k, \lambda\), which can be any positive real numbers. If \(X\) is a random variable with a Burr distribution, given parameter \(\alpha\), this means that \(\alpha\) is acting as one of Burr's parameters, and \(\beta\) and \(\tau\) are the other two. Similarly, understand that a geometric probability mass function has parameter \(p\), where \(p\) is the probability that an event happens, and the geometric distribution models the number of trials needed for the first success. Also, an exponential probability density function models the amount of time that passes before an event occurs, and is defined by parameter \(\beta\).

Apply Distribution properties

(a) Since \(X\) has a conditional Burr distribution and \(\alpha\) has a geometric pmf, the unconditional distribution of \(X\) is still a Burr distribution. This is because the geometric distribution is the only discrete distribution that has the memoryless property, and this property does not change the underlined distribution of given variable. (b) If \(\alpha\) has the exponential pdf, we need to use the property of the exponential distribution to derive the unconditional pdf of \(X\). The property we rely on is that the sum of independent exponential random variables follows a gamma distribution.

Calculate the unconditional pdf

If we adopt the exponential property, the unconditional pdf of \(X\) is given by the gamma distribution. However, as it stands, without specific values to plug into the parameters, we cannot derive a generic formula for the pdf. The unconditional pdf depends on the parameters \(\beta\) and \(\tau\) as well as the functional form of the Burr distribution.

Key concepts

Conditional Distributions

In probability and statistics, conditional distributions are fundamental for understanding the behavior of a random variable given certain conditions or events have occurred. Imagine rolling a dice with the condition that it only counts if the result is even. The distribution of outcomes is different from the original, unrestricted roll. In the case of the Burr distribution in the exercise, the variable's distribution is conditional on parameter \(\alpha\) which, depending on its own distribution, shapes the overall distribution of \(X\).

This concept is especially important when considering random variables in tandem. For example, the height of a randomly chosen person might be dependent on their gender. If we study the height given the person is male, we would consider a conditional distribution.

Probability Mass Function (pmf)

A probability mass function (pmf) pertains to discrete random variables and maps each possible value to its probability. For example, the flip of a fair coin can be modeled with a pmf, where heads (\(H\)) and tails (\(T\)) each have a probability of \(0.5\). With the geometric pmf, we're dealing with the number of trials until the first success in a sequence of Bernoulli trials (like flipping a coin until you get heads), expressed as \( p(1-p)^\alpha \), where \(1-p\) is the probability of failing each individual trial.

Probability Density Function (pdf)

The probability density function (pdf), in contrast to pmf, is used for continuous random variables. It describes the likelihood of a random variable falling within different ranges of values. The integral of the pdf over a certain interval gives the probability that a random variable lies within that range. For instance, the exponential pdf \( \beta^{-1} e^{-\alpha / \beta} \), models time between events in a Poisson process, like the time you wait for a bus at a station. It's crucial to note that the pdf itself is not the probability but rather the density of the probability over an infinitesimally small interval.

Geometric Distribution

The geometric distribution is a discrete distribution that quantifies the number of trials needed for the first success in repeated, independent Bernoulli trials, each with the same probability of success, \(p\). It is memoryless, meaning the probability of success on the next trial is the same, no matter how many failures have previously occurred. This is analogous to constantly flipping a fair coin until it lands on heads, with each toss being independent of previous ones.

Exponential Distribution

The exponential distribution is a continuous probability distribution used to model the time between events in a Poisson process. Key characteristics include being memoryless and having a constant rate of occurrence of events over time. If waiting time for a bus is exponentially distributed with mean \( \beta \), the probability of waiting more than \(t\) minutes is the same as the probability of waiting more than \(t\) minutes, given that you have already waited \(t\) minutes. This unique property is what we refer to as the memoryless property.

Gamma Distribution

The gamma distribution generalizes the exponential distribution for a scenario where the event of interest is the wait time for multiple occurrences rather than one. It encompasses various distributions, such as exponential and chi-squared distributions, as special cases. The gamma distribution is described by two parameters, shape \(k\) and scale \(\theta\), and is particularly helpful for modeling wait times for the \(k^{th}\) event in a Poisson process. If we have several independent exponentially distributed variables, their sum follows a gamma distribution.

Memoryless Property

The memoryless property is a characteristic unique to the exponential and geometric distributions. This property states that the probability of an event occurring in the future is independent of any past events. For instance, no matter how many times you've failed to roll a six with a dice, the probability of rolling a six on the next throw remains the same. The memoryless property simplifies the math behind certain stochastic processes, such as in the exercise where the geometric distribution of parameter \(\alpha\) maintains the overall Burr distribution of the random variable \(X\).