Question

Let \(Y_{1}, \ldots, Y_{k}\) have a Dirichlet distribution with parameters \(\alpha_{1}, \ldots, \alpha_{k}, \alpha_{k+1}\). (a) Show that \(Y_{1}\) has a beta distribution with parameters \(\alpha=\alpha_{1}\) and \(\beta=\alpha_{2}+\) \(\cdots+\alpha_{k+1}\) (b) Show that \(Y_{1}+\cdots+Y_{r}, r \leq k\), has a beta distribution with parameters \(\alpha=\alpha_{1}+\cdots+\alpha_{r}\) and \(\beta=\alpha_{r+1}+\cdots+\alpha_{k+1}\) (c) Show that \(Y_{1}+Y_{2}, Y_{3}+Y_{4}, Y_{5}, \ldots, Y_{k}, k \geq 5\), have a Dirichlet distribution with parameters \(\alpha_{1}+\alpha_{2}, \alpha_{3}+\alpha_{4}, \alpha_{5}, \ldots, \alpha_{k}, \alpha_{k+1}\) Hint: Recall the definition of \(Y_{i}\) in Example \(3.3 .6\) and use the fact that the sum of several independent gamma variables with \(\beta=1\) is a gamma variable.

Short answer

The problem deals with demonstrating whether random variables following a Dirichlet distribution, when in specific formats, follow a Beta or Dirichlet distribution. This demonstration relies on properties of the Dirichlet distribution being a multivariate form of the Beta distribution and the numerical range of independent Gamma random variables. This problem in the end proves that each of the conditions mentioned viz., a component, sum of certain components, and certain pairs sum of the random variables following Dirichlet distribution can be represented by either beta distribution or Dirichlet distribution.

Step-by-step solution

Focus for Part (a)

In this step, consider the case when \(Y_{1}\) follows a Beta distribution. By recalling the Dirichlet distribution as a multivariate form of Beta distribution, a value from a Dirichlet distribution is generated by normalizing a vector of independent Gamma random variables. In other words, for \(Y_{1}\), we have: \[\frac{Gamma(\alpha_{1},1)}{Gamma(\alpha_{1},1)+\ldots+Gamma(\alpha_{k+1},1)}\] Here, the numerator follows a gamma distribution with parameters \(\alpha_{1},1\) and the denominator is the sum of gamma distributions with parameter 1 (which follows another gamma distribution). Hence, it proves that \(Y_{1}\) has a beta distribution.

Focus for Part (b)

Let's consider the case with sum of Dirichlet components \(Y_{1}+\ldots+Y_{r}\), where \(r \leq k\). By following same approach in part (a), note that the numerator now becomes the sum of gamma distributions for \(1,2,\ldots,r\) elements, which according to the hint is another gamma distribution. The denominator stays the same which is the sum of all gamma distributions and hence, it proves that \(Y_{1}+\ldots+Y_{r}\) has a beta distribution.

Focus for Part (c)

Consider the case \(Y_{1}+Y_{2}, Y_{3}+Y_{4}, Y_{5}, \ldots, Y_{k}\). For each of these fractions (two elements summed, single element, and so on), we have the similar argument where numerator is a gamma distribution (sum of few gamma distributions) and denominator is the sum of all gamma distributions. Hence, this proves it follows the Dirichlet distribution with parameters as the sum of respective \(\alpha_{i}\)s.

Key concepts

Beta Distribution

The Beta distribution is a continuous probability distribution defined on the interval \(0, 1\) and is parameterized by two positive shape parameters, often denoted as \(\alpha\) and \(\beta\). It is particularly useful for modeling the distribution of proportions or probabilities, where the outcome is limited to a range between 0 and 1.

Mathematically, the probability density function (PDF) of the Beta distribution is given by the formula: \[ f(x; \alpha, \beta) = \frac{x^{\alpha - 1}(1-x)^{\beta - 1}}{B(\alpha, \beta)} \] where \(B(\alpha, \beta)\) is the beta function, which serves as a normalization constant to ensure that the total probability integrates to 1.

The Beta distribution is related to the Dirichlet distribution, which is a multivariate generalization of the Beta distribution. In the exercise, it's shown that a component of the Dirichlet distribution, \(Y_{1}\), follows a Beta distribution, deriving from the fact that each \(Y_{i}\) is based on independent Gamma random variables that are normalized.

Gamma Distribution

The Gamma distribution is a two-parameter family of continuous probability distributions with wide applications in various fields such as queuing models, meteorology, and finance. The shape parameter \(\alpha\) and the scale parameter \(\theta\) – or alternatively the rate parameter \(\beta = 1/\theta\) – define its shape and scale, respectively.

The PDF of a Gamma distribution is \[ f(x; \alpha, \beta) = \frac{\beta^{\alpha}}{\Gamma(\alpha)}x^{\alpha - 1}e^{-\beta x} \] for \(x > 0\) and \(\alpha, \beta > 0\), where \(\Gamma(\alpha)\) is the gamma function. This distribution becomes relevant in the exercise since a Dirichlet distribution is built from normalized Gamma-distributed variables, and the sums of these independent Gamma variables themselves follow Gamma distributions.

Probability Density Function

A Probability Density Function (PDF) describes the likelihood of a continuous random variable taking on a particular value. The PDF is fundamental in statistics as it can model the distribution of random variables. For any continuous distribution, the area under the PDF across an interval gives the probability that the variable falls within that interval.

In the context of the exercise, we use the PDFs of both the Beta and Gamma distributions to demonstrate and understand the behaviors of \(Y_{1}\) and other variables within the Dirichlet distribution. These functions help to verify that certain random variables indeed follow specific distributions by conforming to the corresponding PDFs.

Statistical Independence

Statistical independence is a critical concept that signifies the absence of any association between two random variables. If two variables are independent, the occurrence of one event does not affect the probability of occurrence of the other.

In probability theory, statistical independence is mathematically represented by showing that the probability of the intersection of two independent events equals the product of their individual probabilities: \[ P(A \cap B) = P(A) \cdot P(B) \] The exercise leverages the property of independence among Gamma random variables to derive the distributions for \(Y_{1}\), \(Y_{1}+ \cdots +Y_{r}\), and other combinations within the Dirichlet distribution. This independence is crucial for the validity of the conclusions drawn about the respective probability distributions of these combinations.