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 \(V_{1}=U_{1}^{1 / \mathrm{a}}\) 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 \(X\).

Short answer

The process of this algorithm starts by generating two independent uniform random variables. These variables are then transformed into new ones by raising to the power of \(1/\alpha\) and \(1/\beta\). The resultants are then summed up. If their sum is less than or equal to 1, \(X\) is computed as the ratio of the transformed \(U_1\) to the sum, otherwise, the process repeats from the beginning. The value of \(X\) is then the output of this process.

Step-by-step solution

Generate \(U_1\) and \(U_2\) with uniform distribution

The first step is to generate two independent random variables (\(U_1\) and \(U_2\)) with uniform distribution over the interval from 0 to 1.

Transformation of \(U_1\) and \(U_2\)

Next, \(U_1\) and \(U_2\) are transformed into new random variables \(V_1\) and \(V_2\) by raising \(U_1\) to the power \(1/\alpha\) and \(U_2\) to \(1/\beta\) respectively. Thus, set \(V_1 = U_1^{1/\alpha}\) and \(V_2 = U_2^{1/\beta}\).

Sum up \(V_1\) and \(V_2\) to get \(W\)

Sum up \(V_1\) and \(V_2\) to calculate \(W\). Therefore, \(W = V_1 + V_2\).

Condition for \(W\)

If \(W \leq 1\), then proceed to generate \(X\), else, return to Step 1.

Calculate \(X\)

If \(W\) is less than or equal to 1, calculate \(X\) as the ratio of \(V_1\) to \(W\). Therefore, \(X = V_1 / W\). The value of \(X\) is the output of the algorithm.

Key concepts

Accept/Reject Algorithm

The Accept/Reject Algorithm is a method used in Monte Carlo simulations, primarily for generating random samples from a probability distribution that might be difficult to sample from directly. Here's the process in simple terms:

• Initially, you measure random variables that follow a known distribution. In this case, uniform distribution is used.
• The algorithm generates potential samples, but they are not immediately accepted.
• These samples undergo a testing process. Only samples that meet certain criteria are accepted.
• If a sample is not accepted, the process starts over with a new set of potential samples.

In the context of the exercise, you generate two uniform random variables, transform them, and then check if their sum is within a specific boundary. If not, you repeat the process. This ensures that the output you finally deliver adheres to the distribution you desire for your simulation.

Uniform Distribution

Uniform distribution is a fundamental concept in probability which describes a condition where all outcomes are equally likely.When we talk about a uniform distribution on the interval from 0 to 1, it means every number between 0 and 1 has the same chance of being chosen.Key properties of uniform distribution:

• The probability density function (PDF) is constant over the specified range.
• The mean of a uniform distribution from 0 to 1 is 0.5.
• The variance is \(\frac{1}{12}\).

In our exercise, the uniform distributions of the variables \(U_1\) and \(U_2\) are essential as a starting point. Because they are iid (independent and identically distributed), the transformations applied to them will still maintain a level of randomness needed for the accept/reject criteria.

Transformation of Variables

Transformation of variables is a technique used to convert one type of random variable into another. This concept is crucial in ensuring that we can work with different types of distributions by transforming variables into ones that are easier to work with or that better suit our needs. In the exercise, we perform transformations on the uniform random variables:

• For \(U_1\), the transformation is done using \(V_1 = U_1^{1/\alpha}\).
• Similarly for \(U_2\), the transformation is \(V_2 = U_2^{1/\beta}\).

These transformations aim to adjust the shape of the original uniform data to more complex distributions. It allows the generated variables to approach the desired distributions needed for the specific simulations, as seen in Monte Carlo methods. Such transformations impact both the expected outcome and its variance, effectively modeling more complex phenomena than the original uniform distribution.