Question

For \(\alpha>0\) and \(\beta>0\), consider the following accept-reject algorithm: 1\. Generate \(U_{1}\) and \(U_{2}\) iid uniform \((0,1)\) random variables. Set \(V_{1}=U_{1}^{1 / \alpha}\) and \(V_{2}=U_{2}^{1 / \beta}\) 2\. Set \(W=V_{1}+V_{2}\). If \(W \leq 1\), set \(X=V_{1} / W\); else go to step 1 . 3\. Deliver \(X\). Show that \(X\) has a beta distribution with parameters \(\alpha\) and \(\beta,(3.3 .9) .\) See Kennedy and Gentle (1980).

Short answer

X, yielded from the given algorithm, does follow the beta distribution with parameters \(\alpha\) and \(\beta\), which can be verified by comparing the derived distribution of X with the probability density function of beta distribution.

Step-by-step solution

Accept-Reject Algorithm

Firstly, the algorithm is followed. \(U_{1}\) and \(U_{2}\) are iid uniform \((0,1)\) random variables. \(V_{1}=U_{1}^{1 / \alpha}\) and \(V_{2}=U_{2}^{1 / \beta}\) are yielded. Then \(W=V_{1}+V_{2}\) is calculated. If \(W \leq 1\), set \(X=V_{1} / W\), or else repeat the process.

Calculate Probability Density Function

The next step is to calculate the joint probability density function of \(V_1\) and \(V_2\). If \(V_1\), \(V_2\) are dependent and both in \([0, 1]\), then the joint density function \(f_{V_1, V_2}(x, y)\) is equal to the product of individual density functions \(f_{V_1}(x) \cdot f_{V_2}(y)\). The sum \(Z = V_1 + V_2\) only takes values between 0 and 2. So the conditional density function of \(V_1\), given \(V_1 + V_2 = z\) is \(f_{V_1|V_1 + V_2 = z}(v)\) when \(0 \leq v \leq z\). After integration over \(z\), the density function of \(X = V_1/(V_1 + V_2)\) is calculated.

Compare with Beta Distribution

The last step is to compare the derived distribution of X with the beta distribution. Beta distribution with parameters \(\alpha\) and \(\beta\) has the density function \(B(\alpha, \beta)x^{\alpha - 1}(1 - x)^{\beta - 1}\), where \(B(\alpha, \beta)\) is the beta function. If these two functions match, then it successfully shows that \(X\) follows a beta distribution with parameters \(\alpha\) and \(\beta\).