Question

Consider the following algorithm: (1) Generate \(U\) and \(V\) independent uniform \((-1,1)\) random variables. (2) Set \(W=U^{2}+V^{2}\). (3) If \(W>1\) goto Step (1). (4) Set \(Z=\sqrt{(-2 \log W) / W}\) and let \(X_{1}=U Z\) and \(X_{2}=V Z\).

Short answer

This algorithm describes a process to generate two independent random variables that follow a standard normal distribution using uniform random variables through a series of calculations involving loops and conditions. \(W\), \(Z\), \(X_{1}\) and \(X_{2}\) are calculated as described in the steps.

Step-by-step solution

Generating U and V

Generate \(U\) and \(V\) which are independent uniform random variables. A uniform random variable is one where all outcomes are equally likely, and in this case, it ranges between -1 and 1.

Evaluate W

Evaluate \(W\) with the formula \(W = U^{2}+V^{2}\). You're basically squaring each of the generated random variables and then adding them together.

W Condition

If \(W>1\), step back to step 1 and generate new random variables for calculation. This is a loop until the condition \(W <= 1\) is met.

Evaluate Z

Evaluate \(Z\) with the formula \(Z=\sqrt{-2 \log W / W}\). This involves taking the negative natural logarithm of \(W\), multiplying it by -2, and then dividing by \(W\). Following this, take the square root of the entire expression.

Generate X1 and X2

Finally, generate \(X_{1}=UZ\) and \(X_{2}=VZ\). For each of these, you are simply multiplying the original uniform random variable by the \(Z\) variable calculated in the previous step.

Key concepts

Random Variables

Random variables are fundamental to understanding probability and statistics. In simple terms, a random variable is a numerical description of the outcome of a random phenomenon. When performing Monte Carlo simulations or any probabilistic experiment, random variables play a central role in modeling various scenarios or events.

To give you a better perspective, random variables can either be discrete or continuous:

• Discrete random variables take on a countable number of values. An example might be the roll of a dice, which can result in one of six possible outcomes.
• Continuous random variables, on the other hand, can take on an infinite number of values within a given range, such as the weight or height of a person.

In our exercise, we focus on continuous random variables, specifically uniform random variables, which are pivotal in random number generation techniques for Monte Carlo simulations.

Uniform Distribution

Uniform distribution is a type of probability distribution in which all outcomes are equally likely within a certain range. In specific applications like the one described in the exercise, we use a uniform distribution to generate random variables.

For instance, when a random variable is uniformly distributed over the interval (-1, 1), this means that any number within this interval is equally likely to be chosen. Mathematically, if a random variable \(U\) follows a uniform distribution in the interval \(a\) to \(b\), its probability density function (PDF) is given as:\[ f(x) = \frac{1}{b-a}, \quad \text{for } a \leq x \leq b\]
This uniform setting provides a straightforward method to avoid biases in simulation, thus useful in generating \(U\) and \(V\) used in the algorithm. Understanding these concepts is necessary to grasp how we derive outcomes that closely mimic real-world randomness.

Box-Muller Transform

The Box-Muller Transform is a powerful mathematical technique used to transform uniform random variables into standard normal (Gaussian) random variables. This method is particularly valuable when simulating data that must adhere to a normal distribution, which is a common assumption in many statistical models.

The exercise introduces a twist on this transform where we utilize the generated uniform variables \(U\) and \(V\) to compute two independent standard normal variables \(X_1\) and \(X_2\). The transform steps in the exercise highlight how to perform these calculations in a sequence:

• First, compute \(W = U^2 + V^2\) and ensure it's less than or equal to 1. This ensures the variables fall within a unit circle, an essential part of the transform's validity.
• Then, use the formula for \(Z = \sqrt{-2 \log(W) / W}\), which involves logarithmic and square root transformations crucial for conforming to the desired Gaussian distribution.
• Finally, the outputs \(X_1 = UZ\) and \(X_2 = VZ\) become well-distributed Gaussian variables ready for statistical analyses.

With the Box-Muller Transform, converting uniform random numbers to Gaussian random numbers becomes efficient, allowing precise modeling in simulations and various analytical tools.