Question

Consider the following mixed discrete-continuous pdf for a random vector \((X, Y)\) (discussed in Casella and George, 1992): $$ f(x, y) \propto\left\\{\begin{array}{ll} \left(\begin{array}{l} n \\ x \end{array}\right) y^{x+\alpha-1}(1-y)^{n-x+\beta-1} & x=0,1, \ldots, n, 00\) and \(\beta>0\). (a) Show that this function is indeed a joint, mixed discrete-continuous pdf by finding the proper constant of proportionality. (b) Determine the conditional pdfs \(f(x \mid y)\) and \(f(y \mid x)\). (c) Write the Gibbs sampler algorithm to generate random samples on \(X\) and \(Y\). (d) Determine the marginal distributions of \(X\) and \(Y\).

Short answer

The joint pdf is a mixed discrete-continuous pdf with the proper constant of proportionality found by applying integration over a certain range for x and y. The conditional pdfs can be calculated using Bayes' theorem and the marginal pdfs are calculated through integration/summation over the other variable. The Gibbs sampler algorithm for generating random samples on a mixed discrete-continuous pdf is based on iterative sampling steps.

Step-by-step solution

Show this is a mixed discrete-continuous pdf

To demonstrate \(f(x, y)\) is a joint, mixed discrete-continuous pdf, integrate \(f(x, y)\) over \(x\) for \(0 \leq x \leq n\) and \(y\) for \(0 < y < 1\). After integrating, the result should be multiplied by an unknown constant \(C\) for normalization. This process will lead to finding the proper constant of proportionality.

Compute the conditional pdfs

Using Bayes' theorem, the conditional pdf \(f(x|y)\) can be computed as the joint pdf \(f(x, y)\) divided by the marginal pdf of \(Y\), \(f(y)\). Likewise, the conditional pdf \(f(y|x)\) can be computed as the joint pdf divided by the marginal pdf of \(X\), \(f(x)\). Here, \(f(x)\) and \(f(y)\) are the marginal pdfs obtained from the first step.

Write the Gibbs sampler algorithm

The Gibbs sampler algorithm proceeds by sampling one variable at a time with the remaining variable(s) fixed to their current values. First, initialize x and y. Next, generate \(x_{t+1}|y=y_t\) using the conditional pdf \(f(x|y)\), and then generate \(y_{t+1}|x=x_{t+1}\) using \(f(y|x)\). Repeat these steps for a large number of iterations.

Determine the marginal distributions

The marginal distribution of a variable in a pdf involves integrating or summing over all other variables. The marginal distribution \(f(x)\) can be obtained by integrating \(f(x, y)\) over \(y\) for all values of \(x\). Similarly, \(f(y)\) can be obtained by summing \(f(x, y)\) over \(x\) for all values of \(y\).